Given any matrices A, B E Mm,n(K), and C, DE Mn,p(K), for all λ € K, we have

College Algebra
1st Edition
ISBN:9781938168383
Author:Jay Abramson
Publisher:Jay Abramson
Chapter7: Systems Of Equations And Inequalities
Section7.7: Solving Systems With Inverses
Problem 1SE: In a previous section, we showed that matrix multiplication is not commutative, that is, ABBA in...
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Proposition 4.1. (1) Given any matrices A € Mm,n(K), B € Mn,p(K), and C € Mp,q(K),
we have
(AB)C= A(BC);
that is, matrix multiplication is associative.
(2) Given any matrices A, B E Mm,n(K), and C, D e Mn,p(K), for all λ = K, we have
(A + B)C = AC + BC
A(C + D) = AC + AD
(XA)C = X(AC)
A(XC) = X(AC),
so that matrix multiplication : Mm,n(K) × Mn,p(K) → Mm,p(K) is bilinear.
Transcribed Image Text:Proposition 4.1. (1) Given any matrices A € Mm,n(K), B € Mn,p(K), and C € Mp,q(K), we have (AB)C= A(BC); that is, matrix multiplication is associative. (2) Given any matrices A, B E Mm,n(K), and C, D e Mn,p(K), for all λ = K, we have (A + B)C = AC + BC A(C + D) = AC + AD (XA)C = X(AC) A(XC) = X(AC), so that matrix multiplication : Mm,n(K) × Mn,p(K) → Mm,p(K) is bilinear.
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