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Theorem List for Metamath Proof Explorer - 22401-22500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremscmateALT 22401* Alternate proof of scmate 22399: An entry of an 𝑁 x 𝑁 scalar matrix over the ring 𝑅. This prove makes use of scmatmats 22400 but is longer and requires more distinct variables. (Contributed by AV, 19-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘† = (𝑁 ScMat 𝑅)    &   πΎ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) β†’ βˆƒπ‘ ∈ 𝐾 (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 ))
 
Theoremscmatscm 22402* The multiplication of a matrix with a scalar matrix corresponds to a scalar multiplication. (Contributed by AV, 28-Dec-2019.)
𝐾 = (Baseβ€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π΄)    &    Γ— = (.rβ€˜π΄)    &   π‘† = (𝑁 ScMat 𝑅)    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐢 ∈ 𝑆) β†’ βˆƒπ‘ ∈ 𝐾 βˆ€π‘š ∈ 𝐡 (𝐢 Γ— π‘š) = (𝑐 βˆ— π‘š))
 
Theoremscmatid 22403 The identity matrix is a scalar matrix. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 18-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   πΈ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   π‘† = (𝑁 ScMat 𝑅)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ (1rβ€˜π΄) ∈ 𝑆)
 
Theoremscmatdmat 22404 A scalar matrix is a diagonal matrix. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 19-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   πΈ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   π‘† = (𝑁 ScMat 𝑅)    &   π· = (𝑁 DMat 𝑅)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ (𝑀 ∈ 𝑆 β†’ 𝑀 ∈ 𝐷))
 
Theoremscmataddcl 22405 The sum of two scalar matrices is a scalar matrix. (Contributed by AV, 25-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   πΈ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   π‘† = (𝑁 ScMat 𝑅)    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆)) β†’ (𝑋(+gβ€˜π΄)π‘Œ) ∈ 𝑆)
 
Theoremscmatsubcl 22406 The difference of two scalar matrices is a scalar matrix. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 19-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   πΈ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   π‘† = (𝑁 ScMat 𝑅)    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆)) β†’ (𝑋(-gβ€˜π΄)π‘Œ) ∈ 𝑆)
 
Theoremscmatmulcl 22407 The product of two scalar matrices is a scalar matrix. (Contributed by AV, 21-Aug-2019.) (Revised by AV, 19-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   πΈ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   π‘† = (𝑁 ScMat 𝑅)    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆)) β†’ (𝑋(.rβ€˜π΄)π‘Œ) ∈ 𝑆)
 
Theoremscmatsgrp 22408 The set of scalar matrices is a subgroup of the matrix group/algebra. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 19-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   πΈ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   π‘† = (𝑁 ScMat 𝑅)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ 𝑆 ∈ (SubGrpβ€˜π΄))
 
Theoremscmatsrng 22409 The set of scalar matrices is a subring of the matrix ring/algebra. (Contributed by AV, 21-Aug-2019.) (Revised by AV, 19-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   πΈ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   π‘† = (𝑁 ScMat 𝑅)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ 𝑆 ∈ (SubRingβ€˜π΄))
 
Theoremscmatcrng 22410 The subring of scalar matrices (over a commutative ring) is a commutative ring. (Contributed by AV, 21-Aug-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   πΈ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   π‘† = (𝑁 ScMat 𝑅)    &   πΆ = (𝐴 β†Ύs 𝑆)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) β†’ 𝐢 ∈ CRing)
 
Theoremscmatsgrp1 22411 The set of scalar matrices is a subgroup of the group/ring of diagonal matrices. (Contributed by AV, 21-Aug-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   πΈ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   π‘† = (𝑁 ScMat 𝑅)    &   π· = (𝑁 DMat 𝑅)    &   πΆ = (𝐴 β†Ύs 𝐷)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ 𝑆 ∈ (SubGrpβ€˜πΆ))
 
Theoremscmatsrng1 22412 The set of scalar matrices is a subring of the ring of diagonal matrices. (Contributed by AV, 21-Aug-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   πΈ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   π‘† = (𝑁 ScMat 𝑅)    &   π· = (𝑁 DMat 𝑅)    &   πΆ = (𝐴 β†Ύs 𝐷)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ 𝑆 ∈ (SubRingβ€˜πΆ))
 
Theoremsmatvscl 22413 Closure of the scalar multiplication in the ring of scalar matrices. (matvscl 22320 analog.) (Contributed by AV, 24-Dec-2019.)
𝐾 = (Baseβ€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &   π‘† = (𝑁 ScMat 𝑅)    &    βˆ— = ( ·𝑠 β€˜π΄)    β‡’   (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐢 ∈ 𝐾 ∧ 𝑋 ∈ 𝑆)) β†’ (𝐢 βˆ— 𝑋) ∈ 𝑆)
 
Theoremscmatlss 22414 The set of scalar matrices is a linear subspace of the matrix algebra. (Contributed by AV, 25-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π‘† = (𝑁 ScMat 𝑅)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ 𝑆 ∈ (LSubSpβ€˜π΄))
 
Theoremscmatstrbas 22415 The set of scalar matrices is the base set of the ring of corresponding scalar matrices. (Contributed by AV, 26-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   πΆ = (𝑁 ScMat 𝑅)    &   π‘† = (𝐴 β†Ύs 𝐢)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ (Baseβ€˜π‘†) = 𝐢)
 
Theoremscmatrhmval 22416* The value of the ring homomorphism 𝐹. (Contributed by AV, 22-Dec-2019.)
𝐾 = (Baseβ€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &    1 = (1rβ€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π΄)    &   πΉ = (π‘₯ ∈ 𝐾 ↦ (π‘₯ βˆ— 1 ))    β‡’   ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐾) β†’ (πΉβ€˜π‘‹) = (𝑋 βˆ— 1 ))
 
Theoremscmatrhmcl 22417* The value of the ring homomorphism 𝐹 is a scalar matrix. (Contributed by AV, 22-Dec-2019.)
𝐾 = (Baseβ€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &    1 = (1rβ€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π΄)    &   πΉ = (π‘₯ ∈ 𝐾 ↦ (π‘₯ βˆ— 1 ))    &   πΆ = (𝑁 ScMat 𝑅)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐾) β†’ (πΉβ€˜π‘‹) ∈ 𝐢)
 
Theoremscmatf 22418* There is a function from a ring to any ring of scalar matrices over this ring. (Contributed by AV, 25-Dec-2019.)
𝐾 = (Baseβ€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &    1 = (1rβ€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π΄)    &   πΉ = (π‘₯ ∈ 𝐾 ↦ (π‘₯ βˆ— 1 ))    &   πΆ = (𝑁 ScMat 𝑅)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ 𝐹:𝐾⟢𝐢)
 
Theoremscmatfo 22419* There is a function from a ring onto any ring of scalar matrices over this ring. (Contributed by AV, 26-Dec-2019.)
𝐾 = (Baseβ€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &    1 = (1rβ€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π΄)    &   πΉ = (π‘₯ ∈ 𝐾 ↦ (π‘₯ βˆ— 1 ))    &   πΆ = (𝑁 ScMat 𝑅)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ 𝐹:𝐾–onto→𝐢)
 
Theoremscmatf1 22420* There is a 1-1 function from a ring to any ring of scalar matrices with positive dimension over this ring. (Contributed by AV, 25-Dec-2019.)
𝐾 = (Baseβ€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &    1 = (1rβ€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π΄)    &   πΉ = (π‘₯ ∈ 𝐾 ↦ (π‘₯ βˆ— 1 ))    &   πΆ = (𝑁 ScMat 𝑅)    β‡’   ((𝑁 ∈ Fin ∧ 𝑁 β‰  βˆ… ∧ 𝑅 ∈ Ring) β†’ 𝐹:𝐾–1-1→𝐢)
 
Theoremscmatf1o 22421* There is a bijection between a ring and any ring of scalar matrices with positive dimension over this ring. (Contributed by AV, 26-Dec-2019.)
𝐾 = (Baseβ€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &    1 = (1rβ€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π΄)    &   πΉ = (π‘₯ ∈ 𝐾 ↦ (π‘₯ βˆ— 1 ))    &   πΆ = (𝑁 ScMat 𝑅)    β‡’   ((𝑁 ∈ Fin ∧ 𝑁 β‰  βˆ… ∧ 𝑅 ∈ Ring) β†’ 𝐹:𝐾–1-1-onto→𝐢)
 
Theoremscmatghm 22422* There is a group homomorphism from the additive group of a ring to the additive group of the ring of scalar matrices over this ring. (Contributed by AV, 22-Dec-2019.)
𝐾 = (Baseβ€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &    1 = (1rβ€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π΄)    &   πΉ = (π‘₯ ∈ 𝐾 ↦ (π‘₯ βˆ— 1 ))    &   πΆ = (𝑁 ScMat 𝑅)    &   π‘† = (𝐴 β†Ύs 𝐢)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ 𝐹 ∈ (𝑅 GrpHom 𝑆))
 
Theoremscmatmhm 22423* There is a monoid homomorphism from the multiplicative group of a ring to the multiplicative group of the ring of scalar matrices over this ring. (Contributed by AV, 29-Dec-2019.)
𝐾 = (Baseβ€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &    1 = (1rβ€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π΄)    &   πΉ = (π‘₯ ∈ 𝐾 ↦ (π‘₯ βˆ— 1 ))    &   πΆ = (𝑁 ScMat 𝑅)    &   π‘† = (𝐴 β†Ύs 𝐢)    &   π‘€ = (mulGrpβ€˜π‘…)    &   π‘‡ = (mulGrpβ€˜π‘†)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ 𝐹 ∈ (𝑀 MndHom 𝑇))
 
Theoremscmatrhm 22424* There is a ring homomorphism from a ring to the ring of scalar matrices over this ring. (Contributed by AV, 29-Dec-2019.)
𝐾 = (Baseβ€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &    1 = (1rβ€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π΄)    &   πΉ = (π‘₯ ∈ 𝐾 ↦ (π‘₯ βˆ— 1 ))    &   πΆ = (𝑁 ScMat 𝑅)    &   π‘† = (𝐴 β†Ύs 𝐢)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) β†’ 𝐹 ∈ (𝑅 RingHom 𝑆))
 
Theoremscmatrngiso 22425* There is a ring isomorphism from a ring to the ring of scalar matrices over this ring with positive dimension. (Contributed by AV, 29-Dec-2019.)
𝐾 = (Baseβ€˜π‘…)    &   π΄ = (𝑁 Mat 𝑅)    &    1 = (1rβ€˜π΄)    &    βˆ— = ( ·𝑠 β€˜π΄)    &   πΉ = (π‘₯ ∈ 𝐾 ↦ (π‘₯ βˆ— 1 ))    &   πΆ = (𝑁 ScMat 𝑅)    &   π‘† = (𝐴 β†Ύs 𝐢)    β‡’   ((𝑁 ∈ Fin ∧ 𝑁 β‰  βˆ… ∧ 𝑅 ∈ Ring) β†’ 𝐹 ∈ (𝑅 RingIso 𝑆))
 
Theoremscmatric 22426 A ring is isomorphic to every ring of scalar matrices over this ring with positive dimension. (Contributed by AV, 29-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   πΆ = (𝑁 ScMat 𝑅)    &   π‘† = (𝐴 β†Ύs 𝐢)    β‡’   ((𝑁 ∈ Fin ∧ 𝑁 β‰  βˆ… ∧ 𝑅 ∈ Ring) β†’ 𝑅 β‰ƒπ‘Ÿ 𝑆)
 
Theoremmat0scmat 22427 The empty matrix over a ring is a scalar matrix (and therefore, by scmatdmat 22404, also a diagonal matrix). (Contributed by AV, 21-Dec-2019.)
(𝑅 ∈ Ring β†’ βˆ… ∈ (βˆ… ScMat 𝑅))
 
Theoremmat1scmat 22428 A 1-dimensional matrix over a ring is always a scalar matrix (and therefore, by scmatdmat 22404, also a diagonal matrix). (Contributed by AV, 21-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    β‡’   ((𝑁 ∈ 𝑉 ∧ (β™―β€˜π‘) = 1 ∧ 𝑅 ∈ Ring) β†’ (𝑀 ∈ 𝐡 β†’ 𝑀 ∈ (𝑁 ScMat 𝑅)))
 
11.4.6  Multiplication of a matrix with a "column vector"

The module of 𝑛-dimensional "column vectors" over a ring π‘Ÿ is the 𝑛-dimensional free module over a ring π‘Ÿ, which is the product of 𝑛 -many copies of the ring with componentwise addition and multiplication. Although a "column vector" could also be defined as n x 1 -matrix (according to Wikipedia "Row and column vectors", 22-Feb-2019, https://en.wikipedia.org/wiki/Row_and_column_vectors: "In linear algebra, a column vector [... ] is an m x 1 matrix, that is, a matrix consisting of a single column of m elements"), which would allow for using the matrix multiplication df-mamu 22273 for multiplying a matrix with a column vector, it seems more natural to use the definition of a free (left) module, avoiding to provide a singleton as 1-dimensional index set for the column, and to introduce a new operator df-mvmul 22430 for the multiplication of a matrix with a column vector. In most cases, it is sufficient to regard members of ((Baseβ€˜π‘…) ↑m 𝑁) as "column vectors", because ((Baseβ€˜π‘…) ↑m 𝑁) is the base set of (𝑅 freeLMod 𝑁), see frlmbasmap 21680. See also the statements in [Lang] p. 508.

 
Syntaxcmvmul 22429 Syntax for the operator for the multiplication of a vector with a matrix.
class maVecMul
 
Definitiondf-mvmul 22430* The operator which multiplies an M x N -matrix with an N-dimensional vector. (Contributed by AV, 23-Feb-2019.)
maVecMul = (π‘Ÿ ∈ V, π‘œ ∈ V ↦ ⦋(1st β€˜π‘œ) / π‘šβ¦Œβ¦‹(2nd β€˜π‘œ) / π‘›β¦Œ(π‘₯ ∈ ((Baseβ€˜π‘Ÿ) ↑m (π‘š Γ— 𝑛)), 𝑦 ∈ ((Baseβ€˜π‘Ÿ) ↑m 𝑛) ↦ (𝑖 ∈ π‘š ↦ (π‘Ÿ Ξ£g (𝑗 ∈ 𝑛 ↦ ((𝑖π‘₯𝑗)(.rβ€˜π‘Ÿ)(π‘¦β€˜π‘—)))))))
 
Theoremmvmulfval 22431* Functional value of the matrix vector multiplication operator. (Contributed by AV, 23-Feb-2019.)
Γ— = (𝑅 maVecMul βŸ¨π‘€, π‘βŸ©)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ 𝑉)    &   (πœ‘ β†’ 𝑀 ∈ Fin)    &   (πœ‘ β†’ 𝑁 ∈ Fin)    β‡’   (πœ‘ β†’ Γ— = (π‘₯ ∈ (𝐡 ↑m (𝑀 Γ— 𝑁)), 𝑦 ∈ (𝐡 ↑m 𝑁) ↦ (𝑖 ∈ 𝑀 ↦ (𝑅 Ξ£g (𝑗 ∈ 𝑁 ↦ ((𝑖π‘₯𝑗) Β· (π‘¦β€˜π‘—)))))))
 
Theoremmvmulval 22432* Multiplication of a vector with a matrix. (Contributed by AV, 23-Feb-2019.)
Γ— = (𝑅 maVecMul βŸ¨π‘€, π‘βŸ©)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ 𝑉)    &   (πœ‘ β†’ 𝑀 ∈ Fin)    &   (πœ‘ β†’ 𝑁 ∈ Fin)    &   (πœ‘ β†’ 𝑋 ∈ (𝐡 ↑m (𝑀 Γ— 𝑁)))    &   (πœ‘ β†’ π‘Œ ∈ (𝐡 ↑m 𝑁))    β‡’   (πœ‘ β†’ (𝑋 Γ— π‘Œ) = (𝑖 ∈ 𝑀 ↦ (𝑅 Ξ£g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) Β· (π‘Œβ€˜π‘—))))))
 
Theoremmvmulfv 22433* A cell/element in the vector resulting from a multiplication of a vector with a matrix. (Contributed by AV, 23-Feb-2019.)
Γ— = (𝑅 maVecMul βŸ¨π‘€, π‘βŸ©)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ 𝑉)    &   (πœ‘ β†’ 𝑀 ∈ Fin)    &   (πœ‘ β†’ 𝑁 ∈ Fin)    &   (πœ‘ β†’ 𝑋 ∈ (𝐡 ↑m (𝑀 Γ— 𝑁)))    &   (πœ‘ β†’ π‘Œ ∈ (𝐡 ↑m 𝑁))    &   (πœ‘ β†’ 𝐼 ∈ 𝑀)    β‡’   (πœ‘ β†’ ((𝑋 Γ— π‘Œ)β€˜πΌ) = (𝑅 Ξ£g (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) Β· (π‘Œβ€˜π‘—)))))
 
Theoremmavmulval 22434* Multiplication of a vector with a square matrix. (Contributed by AV, 23-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &    Γ— = (𝑅 maVecMul βŸ¨π‘, π‘βŸ©)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ 𝑉)    &   (πœ‘ β†’ 𝑁 ∈ Fin)    &   (πœ‘ β†’ 𝑋 ∈ (Baseβ€˜π΄))    &   (πœ‘ β†’ π‘Œ ∈ (𝐡 ↑m 𝑁))    β‡’   (πœ‘ β†’ (𝑋 Γ— π‘Œ) = (𝑖 ∈ 𝑁 ↦ (𝑅 Ξ£g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) Β· (π‘Œβ€˜π‘—))))))
 
Theoremmavmulfv 22435* A cell/element in the vector resulting from a multiplication of a vector with a square matrix. (Contributed by AV, 6-Dec-2018.) (Revised by AV, 18-Feb-2019.) (Revised by AV, 23-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &    Γ— = (𝑅 maVecMul βŸ¨π‘, π‘βŸ©)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ 𝑉)    &   (πœ‘ β†’ 𝑁 ∈ Fin)    &   (πœ‘ β†’ 𝑋 ∈ (Baseβ€˜π΄))    &   (πœ‘ β†’ π‘Œ ∈ (𝐡 ↑m 𝑁))    &   (πœ‘ β†’ 𝐼 ∈ 𝑁)    β‡’   (πœ‘ β†’ ((𝑋 Γ— π‘Œ)β€˜πΌ) = (𝑅 Ξ£g (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) Β· (π‘Œβ€˜π‘—)))))
 
Theoremmavmulcl 22436 Multiplication of an NxN matrix with an N-dimensional vector results in an N-dimensional vector. (Contributed by AV, 6-Dec-2018.) (Revised by AV, 23-Feb-2019.) (Proof shortened by AV, 23-Jul-2019.)
𝐴 = (𝑁 Mat 𝑅)    &    Γ— = (𝑅 maVecMul βŸ¨π‘, π‘βŸ©)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑁 ∈ Fin)    &   (πœ‘ β†’ 𝑋 ∈ (Baseβ€˜π΄))    &   (πœ‘ β†’ π‘Œ ∈ (𝐡 ↑m 𝑁))    β‡’   (πœ‘ β†’ (𝑋 Γ— π‘Œ) ∈ (𝐡 ↑m 𝑁))
 
Theorem1mavmul 22437 Multiplication of the identity NxN matrix with an N-dimensional vector results in the vector itself. (Contributed by AV, 9-Feb-2019.) (Revised by AV, 23-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (𝑅 maVecMul βŸ¨π‘, π‘βŸ©)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑁 ∈ Fin)    &   (πœ‘ β†’ π‘Œ ∈ (𝐡 ↑m 𝑁))    β‡’   (πœ‘ β†’ ((1rβ€˜π΄) Β· π‘Œ) = π‘Œ)
 
Theoremmavmulass 22438 Associativity of the multiplication of two NxN matrices with an N-dimensional vector. (Contributed by AV, 9-Feb-2019.) (Revised by AV, 25-Feb-2019.) (Proof shortened by AV, 22-Jul-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (𝑅 maVecMul βŸ¨π‘, π‘βŸ©)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑁 ∈ Fin)    &   (πœ‘ β†’ π‘Œ ∈ (𝐡 ↑m 𝑁))    &    Γ— = (𝑅 maMul βŸ¨π‘, 𝑁, π‘βŸ©)    &   (πœ‘ β†’ 𝑋 ∈ (Baseβ€˜π΄))    &   (πœ‘ β†’ 𝑍 ∈ (Baseβ€˜π΄))    β‡’   (πœ‘ β†’ ((𝑋 Γ— 𝑍) Β· π‘Œ) = (𝑋 Β· (𝑍 Β· π‘Œ)))
 
Theoremmavmuldm 22439 The domain of the matrix vector multiplication function. (Contributed by AV, 27-Feb-2019.)
𝐡 = (Baseβ€˜π‘…)    &   πΆ = (𝐡 ↑m (𝑀 Γ— 𝑁))    &   π· = (𝐡 ↑m 𝑁)    &    Β· = (𝑅 maVecMul βŸ¨π‘€, π‘βŸ©)    β‡’   ((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ Fin ∧ 𝑁 ∈ Fin) β†’ dom Β· = (𝐢 Γ— 𝐷))
 
Theoremmavmulsolcl 22440 Every solution of the equation π΄βˆ—π‘‹ = π‘Œ for a matrix 𝐴 and a vector 𝐡 is a vector. (Contributed by AV, 27-Feb-2019.)
𝐡 = (Baseβ€˜π‘…)    &   πΆ = (𝐡 ↑m (𝑀 Γ— 𝑁))    &   π· = (𝐡 ↑m 𝑁)    &    Β· = (𝑅 maVecMul βŸ¨π‘€, π‘βŸ©)    &   πΈ = (𝐡 ↑m 𝑀)    β‡’   (((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑀 β‰  βˆ…) ∧ (𝑅 ∈ 𝑉 ∧ π‘Œ ∈ 𝐸)) β†’ ((𝐴 Β· 𝑋) = π‘Œ β†’ 𝑋 ∈ 𝐷))
 
Theoremmavmul0 22441 Multiplication of a 0-dimensional matrix with a 0-dimensional vector. (Contributed by AV, 28-Feb-2019.)
Β· = (𝑅 maVecMul βŸ¨π‘, π‘βŸ©)    β‡’   ((𝑁 = βˆ… ∧ 𝑅 ∈ 𝑉) β†’ (βˆ… Β· βˆ…) = βˆ…)
 
Theoremmavmul0g 22442 The result of the 0-dimensional multiplication of a matrix with a vector is always the empty set. (Contributed by AV, 1-Mar-2019.)
Β· = (𝑅 maVecMul βŸ¨π‘, π‘βŸ©)    β‡’   ((𝑁 = βˆ… ∧ 𝑅 ∈ 𝑉) β†’ (𝑋 Β· π‘Œ) = βˆ…)
 
Theoremmvmumamul1 22443* The multiplication of an MxN matrix with an N-dimensional vector corresponds to the matrix multiplication of an MxN matrix with an Nx1 matrix. (Contributed by AV, 14-Mar-2019.)
Γ— = (𝑅 maMul βŸ¨π‘€, 𝑁, {βˆ…}⟩)    &    Β· = (𝑅 maVecMul βŸ¨π‘€, π‘βŸ©)    &   π΅ = (Baseβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑀 ∈ Fin)    &   (πœ‘ β†’ 𝑁 ∈ Fin)    &   (πœ‘ β†’ 𝐴 ∈ (𝐡 ↑m (𝑀 Γ— 𝑁)))    &   (πœ‘ β†’ π‘Œ ∈ (𝐡 ↑m 𝑁))    &   (πœ‘ β†’ 𝑍 ∈ (𝐡 ↑m (𝑁 Γ— {βˆ…})))    β‡’   (πœ‘ β†’ (βˆ€π‘— ∈ 𝑁 (π‘Œβ€˜π‘—) = (π‘—π‘βˆ…) β†’ βˆ€π‘– ∈ 𝑀 ((𝐴 Β· π‘Œ)β€˜π‘–) = (𝑖(𝐴 Γ— 𝑍)βˆ…)))
 
Theoremmavmumamul1 22444* The multiplication of an NxN matrix with an N-dimensional vector corresponds to the matrix multiplication of an NxN matrix with an Nx1 matrix. (Contributed by AV, 14-Mar-2019.)
𝐴 = (𝑁 Mat 𝑅)    &    Γ— = (𝑅 maMul βŸ¨π‘, 𝑁, {βˆ…}⟩)    &    Β· = (𝑅 maVecMul βŸ¨π‘, π‘βŸ©)    &   π΅ = (Baseβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑁 ∈ Fin)    &   (πœ‘ β†’ 𝑋 ∈ (Baseβ€˜π΄))    &   (πœ‘ β†’ π‘Œ ∈ (𝐡 ↑m 𝑁))    &   (πœ‘ β†’ 𝑍 ∈ (𝐡 ↑m (𝑁 Γ— {βˆ…})))    β‡’   (πœ‘ β†’ (βˆ€π‘— ∈ 𝑁 (π‘Œβ€˜π‘—) = (π‘—π‘βˆ…) β†’ βˆ€π‘– ∈ 𝑁 ((𝑋 Β· π‘Œ)β€˜π‘–) = (𝑖(𝑋 Γ— 𝑍)βˆ…)))
 
11.4.7  Replacement functions for a square matrix
 
Syntaxcmarrep 22445 Syntax for the row replacing function for a square matrix.
class matRRep
 
SyntaxcmatrepV 22446 Syntax for the function replacing a column of a matrix by a vector.
class matRepV
 
Definitiondf-marrep 22447* Define the matrices whose k-th row is replaced by 0's and an arbitrary element of the underlying ring at the l-th column. (Contributed by AV, 12-Feb-2019.)
matRRep = (𝑛 ∈ V, π‘Ÿ ∈ V ↦ (π‘š ∈ (Baseβ€˜(𝑛 Mat π‘Ÿ)), 𝑠 ∈ (Baseβ€˜π‘Ÿ) ↦ (π‘˜ ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑖 = π‘˜, if(𝑗 = 𝑙, 𝑠, (0gβ€˜π‘Ÿ)), (π‘–π‘šπ‘—))))))
 
Definitiondf-marepv 22448* Function replacing a column of a matrix by a vector. (Contributed by AV, 9-Feb-2019.) (Revised by AV, 26-Feb-2019.)
matRepV = (𝑛 ∈ V, π‘Ÿ ∈ V ↦ (π‘š ∈ (Baseβ€˜(𝑛 Mat π‘Ÿ)), 𝑣 ∈ ((Baseβ€˜π‘Ÿ) ↑m 𝑛) ↦ (π‘˜ ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑗 = π‘˜, (π‘£β€˜π‘–), (π‘–π‘šπ‘—))))))
 
Theoremmarrepfval 22449* First substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.) (Proof shortened by AV, 2-Mar-2024.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘„ = (𝑁 matRRep 𝑅)    &    0 = (0gβ€˜π‘…)    β‡’   π‘„ = (π‘š ∈ 𝐡, 𝑠 ∈ (Baseβ€˜π‘…) ↦ (π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = π‘˜, if(𝑗 = 𝑙, 𝑠, 0 ), (π‘–π‘šπ‘—)))))
 
Theoremmarrepval0 22450* Second substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘„ = (𝑁 matRRep 𝑅)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑀 ∈ 𝐡 ∧ 𝑆 ∈ (Baseβ€˜π‘…)) β†’ (𝑀𝑄𝑆) = (π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = π‘˜, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))))
 
Theoremmarrepval 22451* Third substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘„ = (𝑁 matRRep 𝑅)    &    0 = (0gβ€˜π‘…)    β‡’   (((𝑀 ∈ 𝐡 ∧ 𝑆 ∈ (Baseβ€˜π‘…)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) β†’ (𝐾(𝑀𝑄𝑆)𝐿) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗))))
 
Theoremmarrepeval 22452 An entry of a matrix with a replaced row. (Contributed by AV, 12-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘„ = (𝑁 matRRep 𝑅)    &    0 = (0gβ€˜π‘…)    β‡’   (((𝑀 ∈ 𝐡 ∧ 𝑆 ∈ (Baseβ€˜π‘…)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) β†’ (𝐼(𝐾(𝑀𝑄𝑆)𝐿)𝐽) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 𝑆, 0 ), (𝐼𝑀𝐽)))
 
Theoremmarrepcl 22453 Closure of the row replacement function for square matrices. (Contributed by AV, 13-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    β‡’   (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐡 ∧ 𝑆 ∈ (Baseβ€˜π‘…)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) β†’ (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐿) ∈ 𝐡)
 
Theoremmarepvfval 22454* First substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.) (Proof shortened by AV, 2-Mar-2024.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘„ = (𝑁 matRepV 𝑅)    &   π‘‰ = ((Baseβ€˜π‘…) ↑m 𝑁)    β‡’   π‘„ = (π‘š ∈ 𝐡, 𝑣 ∈ 𝑉 ↦ (π‘˜ ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = π‘˜, (π‘£β€˜π‘–), (π‘–π‘šπ‘—)))))
 
Theoremmarepvval0 22455* Second substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘„ = (𝑁 matRepV 𝑅)    &   π‘‰ = ((Baseβ€˜π‘…) ↑m 𝑁)    β‡’   ((𝑀 ∈ 𝐡 ∧ 𝐢 ∈ 𝑉) β†’ (𝑀𝑄𝐢) = (π‘˜ ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = π‘˜, (πΆβ€˜π‘–), (𝑖𝑀𝑗)))))
 
Theoremmarepvval 22456* Third substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘„ = (𝑁 matRepV 𝑅)    &   π‘‰ = ((Baseβ€˜π‘…) ↑m 𝑁)    β‡’   ((𝑀 ∈ 𝐡 ∧ 𝐢 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) β†’ ((𝑀𝑄𝐢)β€˜πΎ) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, (πΆβ€˜π‘–), (𝑖𝑀𝑗))))
 
Theoremmarepveval 22457 An entry of a matrix with a replaced column. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘„ = (𝑁 matRepV 𝑅)    &   π‘‰ = ((Baseβ€˜π‘…) ↑m 𝑁)    β‡’   (((𝑀 ∈ 𝐡 ∧ 𝐢 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) β†’ (𝐼((𝑀𝑄𝐢)β€˜πΎ)𝐽) = if(𝐽 = 𝐾, (πΆβ€˜πΌ), (𝐼𝑀𝐽)))
 
Theoremmarepvcl 22458 Closure of the column replacement function for square matrices. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘‰ = ((Baseβ€˜π‘…) ↑m 𝑁)    β‡’   ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐡 ∧ 𝐢 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) β†’ ((𝑀(𝑁 matRepV 𝑅)𝐢)β€˜πΎ) ∈ 𝐡)
 
Theoremma1repvcl 22459 Closure of the column replacement function for identity matrices. (Contributed by AV, 15-Feb-2019.) (Revised by AV, 26-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘‰ = ((Baseβ€˜π‘…) ↑m 𝑁)    &    1 = (1rβ€˜π΄)    β‡’   (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐢 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) β†’ (( 1 (𝑁 matRepV 𝑅)𝐢)β€˜πΎ) ∈ 𝐡)
 
Theoremma1repveval 22460 An entry of an identity matrix with a replaced column. (Contributed by AV, 16-Feb-2019.) (Revised by AV, 26-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘‰ = ((Baseβ€˜π‘…) ↑m 𝑁)    &    1 = (1rβ€˜π΄)    &    0 = (0gβ€˜π‘…)    &   πΈ = (( 1 (𝑁 matRepV 𝑅)𝐢)β€˜πΎ)    β‡’   ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐡 ∧ 𝐢 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) β†’ (𝐼𝐸𝐽) = if(𝐽 = 𝐾, (πΆβ€˜πΌ), if(𝐽 = 𝐼, (1rβ€˜π‘…), 0 )))
 
Theoremmulmarep1el 22461 Element by element multiplication of a matrix with an identity matrix with a column replaced by a vector. (Contributed by AV, 16-Feb-2019.) (Revised by AV, 26-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘‰ = ((Baseβ€˜π‘…) ↑m 𝑁)    &    1 = (1rβ€˜π΄)    &    0 = (0gβ€˜π‘…)    &   πΈ = (( 1 (𝑁 matRepV 𝑅)𝐢)β€˜πΎ)    β‡’   ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐡 ∧ 𝐢 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) β†’ ((𝐼𝑋𝐿)(.rβ€˜π‘…)(𝐿𝐸𝐽)) = if(𝐽 = 𝐾, ((𝐼𝑋𝐿)(.rβ€˜π‘…)(πΆβ€˜πΏ)), if(𝐽 = 𝐿, (𝐼𝑋𝐿), 0 )))
 
Theoremmulmarep1gsum1 22462* The sum of element by element multiplications of a matrix with an identity matrix with a column replaced by a vector. (Contributed by AV, 16-Feb-2019.) (Revised by AV, 26-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘‰ = ((Baseβ€˜π‘…) ↑m 𝑁)    &    1 = (1rβ€˜π΄)    &    0 = (0gβ€˜π‘…)    &   πΈ = (( 1 (𝑁 matRepV 𝑅)𝐢)β€˜πΎ)    β‡’   ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐡 ∧ 𝐢 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐽 β‰  𝐾)) β†’ (𝑅 Ξ£g (𝑙 ∈ 𝑁 ↦ ((𝐼𝑋𝑙)(.rβ€˜π‘…)(𝑙𝐸𝐽)))) = (𝐼𝑋𝐽))
 
Theoremmulmarep1gsum2 22463* The sum of element by element multiplications of a matrix with an identity matrix with a column replaced by a vector. (Contributed by AV, 18-Feb-2019.) (Revised by AV, 26-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘‰ = ((Baseβ€˜π‘…) ↑m 𝑁)    &    1 = (1rβ€˜π΄)    &    0 = (0gβ€˜π‘…)    &   πΈ = (( 1 (𝑁 matRepV 𝑅)𝐢)β€˜πΎ)    &    Γ— = (𝑅 maVecMul βŸ¨π‘, π‘βŸ©)    β‡’   ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐡 ∧ 𝐢 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ (𝑋 Γ— 𝐢) = 𝑍)) β†’ (𝑅 Ξ£g (𝑙 ∈ 𝑁 ↦ ((𝐼𝑋𝑙)(.rβ€˜π‘…)(𝑙𝐸𝐽)))) = if(𝐽 = 𝐾, (π‘β€˜πΌ), (𝐼𝑋𝐽)))
 
Theorem1marepvmarrepid 22464 Replacing the ith row by 0's and the ith component of a (column) vector at the diagonal position for the identity matrix with the ith column replaced by the vector results in the matrix itself. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 27-Feb-2019.)
𝑉 = ((Baseβ€˜π‘…) ↑m 𝑁)    &    1 = (1rβ€˜(𝑁 Mat 𝑅))    &   π‘‹ = (( 1 (𝑁 matRepV 𝑅)𝑍)β€˜πΌ)    β‡’   (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) β†’ (𝐼(𝑋(𝑁 matRRep 𝑅)(π‘β€˜πΌ))𝐼) = 𝑋)
 
11.4.8  Submatrices
 
Syntaxcsubma 22465 Syntax for submatrices of a square matrix.
class subMat
 
Definitiondf-subma 22466* Define the submatrices of a square matrix. A submatrix is obtained by deleting a row and a column of the original matrix. Since the indices of a matrix need not to be sequential integers, it does not matter that there may be gaps in the numbering of the indices for the submatrix. The determinants of such submatrices are called the "minors" of the original matrix. (Contributed by AV, 27-Dec-2018.)
subMat = (𝑛 ∈ V, π‘Ÿ ∈ V ↦ (π‘š ∈ (Baseβ€˜(𝑛 Mat π‘Ÿ)) ↦ (π‘˜ ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ (𝑛 βˆ– {π‘˜}), 𝑗 ∈ (𝑛 βˆ– {𝑙}) ↦ (π‘–π‘šπ‘—)))))
 
Theoremsubmabas 22467* Any subset of the index set of a square matrix defines a submatrix of the matrix. (Contributed by AV, 1-Jan-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    β‡’   ((𝑀 ∈ 𝐡 ∧ 𝐷 βŠ† 𝑁) β†’ (𝑖 ∈ 𝐷, 𝑗 ∈ 𝐷 ↦ (𝑖𝑀𝑗)) ∈ (Baseβ€˜(𝐷 Mat 𝑅)))
 
Theoremsubmafval 22468* First substitution for a submatrix. (Contributed by AV, 28-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   π‘„ = (𝑁 subMat 𝑅)    &   π΅ = (Baseβ€˜π΄)    β‡’   π‘„ = (π‘š ∈ 𝐡 ↦ (π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 βˆ– {π‘˜}), 𝑗 ∈ (𝑁 βˆ– {𝑙}) ↦ (π‘–π‘šπ‘—))))
 
Theoremsubmaval0 22469* Second substitution for a submatrix. (Contributed by AV, 28-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   π‘„ = (𝑁 subMat 𝑅)    &   π΅ = (Baseβ€˜π΄)    β‡’   (𝑀 ∈ 𝐡 β†’ (π‘„β€˜π‘€) = (π‘˜ ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ (𝑁 βˆ– {π‘˜}), 𝑗 ∈ (𝑁 βˆ– {𝑙}) ↦ (𝑖𝑀𝑗))))
 
Theoremsubmaval 22470* Third substitution for a submatrix. (Contributed by AV, 28-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   π‘„ = (𝑁 subMat 𝑅)    &   π΅ = (Baseβ€˜π΄)    β‡’   ((𝑀 ∈ 𝐡 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) β†’ (𝐾(π‘„β€˜π‘€)𝐿) = (𝑖 ∈ (𝑁 βˆ– {𝐾}), 𝑗 ∈ (𝑁 βˆ– {𝐿}) ↦ (𝑖𝑀𝑗)))
 
Theoremsubmaeval 22471 An entry of a submatrix of a square matrix. (Contributed by AV, 28-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   π‘„ = (𝑁 subMat 𝑅)    &   π΅ = (Baseβ€˜π΄)    β‡’   ((𝑀 ∈ 𝐡 ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ (𝑁 βˆ– {𝐾}) ∧ 𝐽 ∈ (𝑁 βˆ– {𝐿}))) β†’ (𝐼(𝐾(π‘„β€˜π‘€)𝐿)𝐽) = (𝐼𝑀𝐽))
 
Theorem1marepvsma1 22472 The submatrix of the identity matrix with the ith column replaced by the vector obtained by removing the ith row and the ith column is an identity matrix. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 27-Feb-2019.)
𝑉 = ((Baseβ€˜π‘…) ↑m 𝑁)    &    1 = (1rβ€˜(𝑁 Mat 𝑅))    &   π‘‹ = (( 1 (𝑁 matRepV 𝑅)𝑍)β€˜πΌ)    β‡’   (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝑍 ∈ 𝑉)) β†’ (𝐼((𝑁 subMat 𝑅)β€˜π‘‹)𝐼) = (1rβ€˜((𝑁 βˆ– {𝐼}) Mat 𝑅)))
 
11.5  The determinant
 
11.5.1  Definition and basic properties
 
Syntaxcmdat 22473 Syntax for the matrix determinant function.
class maDet
 
Definitiondf-mdet 22474* Determinant of a square matrix. This definition is based on Leibniz' Formula (see mdetleib 22476). The properties of the axiomatic definition of a determinant according to [Weierstrass] p. 272 are derived from this definition as theorems: "The determinant function is the unique multilinear, alternating and normalized function from the algebra of square matrices of the same dimension over a commutative ring to this ring". Functionality is shown by mdetf 22484. Multilineary means "linear for each row" - the additivity is shown by mdetrlin 22491, the homogeneity by mdetrsca 22492. Furthermore, it is shown that the determinant function is alternating (see mdetralt 22497) and normalized (see mdet1 22490). Finally, uniqueness is shown by mdetuni 22511. As a consequence, the "determinant of a square matrix" is the function value of the determinant function for this square matrix, see mdetleib 22476. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by SO, 10-Jul-2018.)
maDet = (𝑛 ∈ V, π‘Ÿ ∈ V ↦ (π‘š ∈ (Baseβ€˜(𝑛 Mat π‘Ÿ)) ↦ (π‘Ÿ Ξ£g (𝑝 ∈ (Baseβ€˜(SymGrpβ€˜π‘›)) ↦ ((((β„€RHomβ€˜π‘Ÿ) ∘ (pmSgnβ€˜π‘›))β€˜π‘)(.rβ€˜π‘Ÿ)((mulGrpβ€˜π‘Ÿ) Ξ£g (π‘₯ ∈ 𝑛 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯))))))))
 
Theoremmdetfval 22475* First substitution for the determinant definition. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by SO, 9-Jul-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   π΄ = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Baseβ€˜(SymGrpβ€˜π‘))    &   π‘Œ = (β„€RHomβ€˜π‘…)    &   π‘† = (pmSgnβ€˜π‘)    &    Β· = (.rβ€˜π‘…)    &   π‘ˆ = (mulGrpβ€˜π‘…)    β‡’   π· = (π‘š ∈ 𝐡 ↦ (𝑅 Ξ£g (𝑝 ∈ 𝑃 ↦ (((π‘Œ ∘ 𝑆)β€˜π‘) Β· (π‘ˆ Ξ£g (π‘₯ ∈ 𝑁 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯)))))))
 
Theoremmdetleib 22476* Full substitution of our determinant definition (also known as Leibniz' Formula, expanding by columns). Proposition 4.6 in [Lang] p. 514. (Contributed by Stefan O'Rear, 3-Oct-2015.) (Revised by SO, 9-Jul-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   π΄ = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Baseβ€˜(SymGrpβ€˜π‘))    &   π‘Œ = (β„€RHomβ€˜π‘…)    &   π‘† = (pmSgnβ€˜π‘)    &    Β· = (.rβ€˜π‘…)    &   π‘ˆ = (mulGrpβ€˜π‘…)    β‡’   (𝑀 ∈ 𝐡 β†’ (π·β€˜π‘€) = (𝑅 Ξ£g (𝑝 ∈ 𝑃 ↦ (((π‘Œ ∘ 𝑆)β€˜π‘) Β· (π‘ˆ Ξ£g (π‘₯ ∈ 𝑁 ↦ ((π‘β€˜π‘₯)𝑀π‘₯)))))))
 
Theoremmdetleib2 22477* Leibniz' formula can also be expanded by rows. (Contributed by Stefan O'Rear, 9-Jul-2018.) (Proof shortened by AV, 23-Jul-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   π΄ = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Baseβ€˜(SymGrpβ€˜π‘))    &   π‘Œ = (β„€RHomβ€˜π‘…)    &   π‘† = (pmSgnβ€˜π‘)    &    Β· = (.rβ€˜π‘…)    &   π‘ˆ = (mulGrpβ€˜π‘…)    β‡’   ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ (π·β€˜π‘€) = (𝑅 Ξ£g (𝑝 ∈ 𝑃 ↦ (((π‘Œ ∘ 𝑆)β€˜π‘) Β· (π‘ˆ Ξ£g (π‘₯ ∈ 𝑁 ↦ (π‘₯𝑀(π‘β€˜π‘₯))))))))
 
Theoremnfimdetndef 22478 The determinant is not defined for an infinite matrix. (Contributed by AV, 27-Dec-2018.)
𝐷 = (𝑁 maDet 𝑅)    β‡’   (𝑁 βˆ‰ Fin β†’ 𝐷 = βˆ…)
 
Theoremmdetfval1 22479* First substitution of an alternative determinant definition. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by AV, 27-Dec-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   π΄ = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Baseβ€˜(SymGrpβ€˜π‘))    &   π‘Œ = (β„€RHomβ€˜π‘…)    &   π‘† = (pmSgnβ€˜π‘)    &    Β· = (.rβ€˜π‘…)    &   π‘ˆ = (mulGrpβ€˜π‘…)    β‡’   π· = (π‘š ∈ 𝐡 ↦ (𝑅 Ξ£g (𝑝 ∈ 𝑃 ↦ ((π‘Œβ€˜(π‘†β€˜π‘)) Β· (π‘ˆ Ξ£g (π‘₯ ∈ 𝑁 ↦ ((π‘β€˜π‘₯)π‘šπ‘₯)))))))
 
Theoremmdetleib1 22480* Full substitution of an alternative determinant definition (also known as Leibniz' Formula). (Contributed by Stefan O'Rear, 3-Oct-2015.) (Revised by AV, 26-Dec-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   π΄ = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   π‘ƒ = (Baseβ€˜(SymGrpβ€˜π‘))    &   π‘Œ = (β„€RHomβ€˜π‘…)    &   π‘† = (pmSgnβ€˜π‘)    &    Β· = (.rβ€˜π‘…)    &   π‘ˆ = (mulGrpβ€˜π‘…)    β‡’   (𝑀 ∈ 𝐡 β†’ (π·β€˜π‘€) = (𝑅 Ξ£g (𝑝 ∈ 𝑃 ↦ ((π‘Œβ€˜(π‘†β€˜π‘)) Β· (π‘ˆ Ξ£g (π‘₯ ∈ 𝑁 ↦ ((π‘β€˜π‘₯)𝑀π‘₯)))))))
 
Theoremmdet0pr 22481 The determinant function for 0-dimensional matrices on a given ring is the function mapping the empty set to the unity element of that ring. (Contributed by AV, 28-Feb-2019.)
(𝑅 ∈ Ring β†’ (βˆ… maDet 𝑅) = {βŸ¨βˆ…, (1rβ€˜π‘…)⟩})
 
Theoremmdet0f1o 22482 The determinant function for 0-dimensional matrices on a given ring is a bijection from the singleton containing the empty set (empty matrix) onto the singleton containing the unity element of that ring. (Contributed by AV, 28-Feb-2019.)
(𝑅 ∈ Ring β†’ (βˆ… maDet 𝑅):{βˆ…}–1-1-ontoβ†’{(1rβ€˜π‘…)})
 
Theoremmdet0fv0 22483 The determinant of the empty matrix on a given ring is the unity element of that ring. (Contributed by AV, 28-Feb-2019.)
(𝑅 ∈ Ring β†’ ((βˆ… maDet 𝑅)β€˜βˆ…) = (1rβ€˜π‘…))
 
Theoremmdetf 22484 Functionality of the determinant, see also definition in [Lang] p. 513. (Contributed by Stefan O'Rear, 9-Jul-2018.) (Proof shortened by AV, 23-Jul-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   π΄ = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   πΎ = (Baseβ€˜π‘…)    β‡’   (𝑅 ∈ CRing β†’ 𝐷:𝐡⟢𝐾)
 
Theoremmdetcl 22485 The determinant evaluates to an element of the base ring. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by AV, 7-Feb-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   π΄ = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   πΎ = (Baseβ€˜π‘…)    β‡’   ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ (π·β€˜π‘€) ∈ 𝐾)
 
Theoremm1detdiag 22486 The determinant of a 1-dimensional matrix equals its (single) entry. (Contributed by AV, 6-Aug-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   π΄ = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    β‡’   ((𝑅 ∈ CRing ∧ (𝑁 = {𝐼} ∧ 𝐼 ∈ 𝑉) ∧ 𝑀 ∈ 𝐡) β†’ (π·β€˜π‘€) = (𝐼𝑀𝐼))
 
Theoremmdetdiaglem 22487* Lemma for mdetdiag 22488. Previously part of proof for mdet1 22490. (Contributed by SO, 10-Jul-2018.) (Revised by AV, 17-Aug-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   π΄ = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   πΊ = (mulGrpβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   π» = (Baseβ€˜(SymGrpβ€˜π‘))    &   π‘ = (β„€RHomβ€˜π‘…)    &   π‘† = (pmSgnβ€˜π‘)    &    Β· = (.rβ€˜π‘…)    β‡’   (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐡) ∧ βˆ€π‘– ∈ 𝑁 βˆ€π‘— ∈ 𝑁 (𝑖 β‰  𝑗 β†’ (𝑖𝑀𝑗) = 0 ) ∧ (𝑃 ∈ 𝐻 ∧ 𝑃 β‰  ( I β†Ύ 𝑁))) β†’ (((𝑍 ∘ 𝑆)β€˜π‘ƒ) Β· (𝐺 Ξ£g (π‘˜ ∈ 𝑁 ↦ ((π‘ƒβ€˜π‘˜)π‘€π‘˜)))) = 0 )
 
Theoremmdetdiag 22488* The determinant of a diagonal matrix is the product of the entries in the diagonal. (Contributed by AV, 17-Aug-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   π΄ = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   πΊ = (mulGrpβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑀 ∈ 𝐡) β†’ (βˆ€π‘– ∈ 𝑁 βˆ€π‘— ∈ 𝑁 (𝑖 β‰  𝑗 β†’ (𝑖𝑀𝑗) = 0 ) β†’ (π·β€˜π‘€) = (𝐺 Ξ£g (π‘˜ ∈ 𝑁 ↦ (π‘˜π‘€π‘˜)))))
 
Theoremmdetdiagid 22489* The determinant of a diagonal matrix with identical entries is the power of the entry in the diagonal. (Contributed by AV, 17-Aug-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   π΄ = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   πΊ = (mulGrpβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   πΆ = (Baseβ€˜π‘…)    &    Β· = (.gβ€˜πΊ)    β‡’   (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐡 ∧ 𝑋 ∈ 𝐢)) β†’ (βˆ€π‘– ∈ 𝑁 βˆ€π‘— ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑋, 0 ) β†’ (π·β€˜π‘€) = ((β™―β€˜π‘) Β· 𝑋)))
 
Theoremmdet1 22490 The determinant of the identity matrix is 1, i.e. the determinant function is normalized, see also definition in [Lang] p. 513. (Contributed by SO, 10-Jul-2018.) (Proof shortened by AV, 25-Nov-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   π΄ = (𝑁 Mat 𝑅)    &   πΌ = (1rβ€˜π΄)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) β†’ (π·β€˜πΌ) = 1 )
 
Theoremmdetrlin 22491 The determinant function is additive for each row: The matrices X, Y, Z are identical except for the I's row, and the I's row of the matrix X is the componentwise sum of the I's row of the matrices Y and Z. In this case the determinant of X is the sum of the determinants of Y and Z. (Contributed by SO, 9-Jul-2018.) (Proof shortened by AV, 23-Jul-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   π΄ = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &    + = (+gβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    &   (πœ‘ β†’ 𝐼 ∈ 𝑁)    &   (πœ‘ β†’ (𝑋 β†Ύ ({𝐼} Γ— 𝑁)) = ((π‘Œ β†Ύ ({𝐼} Γ— 𝑁)) ∘f + (𝑍 β†Ύ ({𝐼} Γ— 𝑁))))    &   (πœ‘ β†’ (𝑋 β†Ύ ((𝑁 βˆ– {𝐼}) Γ— 𝑁)) = (π‘Œ β†Ύ ((𝑁 βˆ– {𝐼}) Γ— 𝑁)))    &   (πœ‘ β†’ (𝑋 β†Ύ ((𝑁 βˆ– {𝐼}) Γ— 𝑁)) = (𝑍 β†Ύ ((𝑁 βˆ– {𝐼}) Γ— 𝑁)))    β‡’   (πœ‘ β†’ (π·β€˜π‘‹) = ((π·β€˜π‘Œ) + (π·β€˜π‘)))
 
Theoremmdetrsca 22492 The determinant function is homogeneous for each row: If the matrices 𝑋 and 𝑍 are identical except for the 𝐼-th row, and the 𝐼-th row of the matrix 𝑋 is the componentwise product of the 𝐼-th row of the matrix 𝑍 and the scalar π‘Œ, then the determinant of 𝑋 is the determinant of 𝑍 multiplied by π‘Œ. (Contributed by SO, 9-Jul-2018.) (Proof shortened by AV, 23-Jul-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   π΄ = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &   πΎ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐾)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    &   (πœ‘ β†’ 𝐼 ∈ 𝑁)    &   (πœ‘ β†’ (𝑋 β†Ύ ({𝐼} Γ— 𝑁)) = ((({𝐼} Γ— 𝑁) Γ— {π‘Œ}) ∘f Β· (𝑍 β†Ύ ({𝐼} Γ— 𝑁))))    &   (πœ‘ β†’ (𝑋 β†Ύ ((𝑁 βˆ– {𝐼}) Γ— 𝑁)) = (𝑍 β†Ύ ((𝑁 βˆ– {𝐼}) Γ— 𝑁)))    β‡’   (πœ‘ β†’ (π·β€˜π‘‹) = (π‘Œ Β· (π·β€˜π‘)))
 
Theoremmdetrsca2 22493* The determinant function is homogeneous for each row (matrices are given explicitly by their entries). (Contributed by SO, 16-Jul-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   πΎ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝑁 ∈ Fin)    &   ((πœ‘ ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) β†’ 𝑋 ∈ 𝐾)    &   ((πœ‘ ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) β†’ π‘Œ ∈ 𝐾)    &   (πœ‘ β†’ 𝐹 ∈ 𝐾)    &   (πœ‘ β†’ 𝐼 ∈ 𝑁)    β‡’   (πœ‘ β†’ (π·β€˜(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝐹 Β· 𝑋), π‘Œ))) = (𝐹 Β· (π·β€˜(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, π‘Œ)))))
 
Theoremmdetr0 22494* The determinant of a matrix with a row containing only 0's is 0. (Contributed by SO, 16-Jul-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   πΎ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝑁 ∈ Fin)    &   ((πœ‘ ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) β†’ 𝑋 ∈ 𝐾)    &   (πœ‘ β†’ 𝐼 ∈ 𝑁)    β‡’   (πœ‘ β†’ (π·β€˜(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 0 , 𝑋))) = 0 )
 
Theoremmdet0 22495 The determinant of the zero matrix (of dimension greater 0!) is 0. (Contributed by AV, 17-Aug-2019.) (Revised by AV, 3-Jul-2022.)
𝐷 = (𝑁 maDet 𝑅)    &   π΄ = (𝑁 Mat 𝑅)    &   π‘ = (0gβ€˜π΄)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin ∧ 𝑁 β‰  βˆ…) β†’ (π·β€˜π‘) = 0 )
 
Theoremmdetrlin2 22496* The determinant function is additive for each row (matrices are given explicitly by their entries). (Contributed by SO, 16-Jul-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   πΎ = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝑁 ∈ Fin)    &   ((πœ‘ ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) β†’ 𝑋 ∈ 𝐾)    &   ((πœ‘ ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) β†’ π‘Œ ∈ 𝐾)    &   ((πœ‘ ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) β†’ 𝑍 ∈ 𝐾)    &   (πœ‘ β†’ 𝐼 ∈ 𝑁)    β‡’   (πœ‘ β†’ (π·β€˜(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + π‘Œ), 𝑍))) = ((π·β€˜(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, 𝑍))) + (π·β€˜(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, π‘Œ, 𝑍)))))
 
Theoremmdetralt 22497* The determinant function is alternating regarding rows: if a matrix has two identical rows, its determinant is 0. Corollary 4.9 in [Lang] p. 515. (Contributed by SO, 10-Jul-2018.) (Proof shortened by AV, 23-Jul-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   π΄ = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &    0 = (0gβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ 𝐼 ∈ 𝑁)    &   (πœ‘ β†’ 𝐽 ∈ 𝑁)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   (πœ‘ β†’ βˆ€π‘Ž ∈ 𝑁 (πΌπ‘‹π‘Ž) = (π½π‘‹π‘Ž))    β‡’   (πœ‘ β†’ (π·β€˜π‘‹) = 0 )
 
Theoremmdetralt2 22498* The determinant function is alternating regarding rows (matrix is given explicitly by its entries). (Contributed by SO, 16-Jul-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   πΎ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝑁 ∈ Fin)    &   ((πœ‘ ∧ 𝑗 ∈ 𝑁) β†’ 𝑋 ∈ 𝐾)    &   ((πœ‘ ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) β†’ π‘Œ ∈ 𝐾)    &   (πœ‘ β†’ 𝐼 ∈ 𝑁)    &   (πœ‘ β†’ 𝐽 ∈ 𝑁)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    β‡’   (πœ‘ β†’ (π·β€˜(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, π‘Œ)))) = 0 )
 
Theoremmdetero 22499* The determinant function is multilinear (additive and homogeneous for each row (matrices are given explicitly by their entries). Corollary 4.9 in [Lang] p. 515. (Contributed by SO, 16-Jul-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   πΎ = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝑁 ∈ Fin)    &   ((πœ‘ ∧ 𝑗 ∈ 𝑁) β†’ 𝑋 ∈ 𝐾)    &   ((πœ‘ ∧ 𝑗 ∈ 𝑁) β†’ π‘Œ ∈ 𝐾)    &   ((πœ‘ ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) β†’ 𝑍 ∈ 𝐾)    &   (πœ‘ β†’ π‘Š ∈ 𝐾)    &   (πœ‘ β†’ 𝐼 ∈ 𝑁)    &   (πœ‘ β†’ 𝐽 ∈ 𝑁)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    β‡’   (πœ‘ β†’ (π·β€˜(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, (𝑋 + (π‘Š Β· π‘Œ)), if(𝑖 = 𝐽, π‘Œ, 𝑍)))) = (π·β€˜(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, π‘Œ, 𝑍)))))
 
Theoremmdettpos 22500 Determinant is invariant under transposition. Proposition 4.8 in [Lang] p. 514. (Contributed by Stefan O'Rear, 9-Jul-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   π΄ = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    β‡’   ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ (π·β€˜tpos 𝑀) = (π·β€˜π‘€))
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