Prove the following matrix equation. and show your complete solutions.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.4: Similarity And Diagonalization
Problem 40EQ
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Question
Prove the following matrix equation. and show your complete solutions.
![I.
Properties of Matrix Multiplication
(c) If A, B, and C are matrices of the appropriate sizes, then
C(A + B) = CA+CB.
Proof
(a) Suppose that A is m xn, B is n x p, and C is px q. We shall prove the
result for the special case m = 2, n = 3, p = 4, and q = 3. The general proof is
completely analogous.
Let A = [a], B = [bij], C = [cij], AB = D = [d₁j], BC = E= [eij],
(AB)C = F = [fj], and A(BC) = G = [8ij]. We must show that fij = gij for
all i, j. Now
4
fij = Σdikckj=
k=1
=
4
3
Σ Σairbek) cuj
k=1
r=1
The connector "if and only if" means that both statements are true or both statements are false.
Thus (i) if A+ U = A. then a+u = a: and (ii) if a+u = a. then A+ U = A. See Appendix
C, "Introduction to Proofs."](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc69d24e0-a50c-4d84-80f1-8088020fa4fd%2F9f4a7451-e008-429a-a425-0d88c2dfe374%2F15933tq_processed.png&w=3840&q=75)
Transcribed Image Text:I.
Properties of Matrix Multiplication
(c) If A, B, and C are matrices of the appropriate sizes, then
C(A + B) = CA+CB.
Proof
(a) Suppose that A is m xn, B is n x p, and C is px q. We shall prove the
result for the special case m = 2, n = 3, p = 4, and q = 3. The general proof is
completely analogous.
Let A = [a], B = [bij], C = [cij], AB = D = [d₁j], BC = E= [eij],
(AB)C = F = [fj], and A(BC) = G = [8ij]. We must show that fij = gij for
all i, j. Now
4
fij = Σdikckj=
k=1
=
4
3
Σ Σairbek) cuj
k=1
r=1
The connector "if and only if" means that both statements are true or both statements are false.
Thus (i) if A+ U = A. then a+u = a: and (ii) if a+u = a. then A+ U = A. See Appendix
C, "Introduction to Proofs."
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