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Let A, B, and C be m × n matrices 2.) For each m x n matrix, A, there is a unique m x n matrix D such that A + D = 0 We shall write D as -A, so (2) can be written as A+ (-A) = 0 The matrix -A is called the negative of A. we also note that -A is (-1) A.

Let A, B, and C be m × n matrices 2.) For each m x n matrix, A, there is a unique m x n matrix D such that A + D = 0 We shall write D as -A, so (2) can be written as A+ (-A) = 0 The matrix -A is called the negative of A. we also note that -A is (-1) A.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Prove the following matrix equation. and show your complete solutions.

 

I. Let A, B, and C be m x n matrices 2.) For each m x n matrix, A, there is a unique m x n matrix D such that A + D = 0 Properties of Matrix Addition We shall write D as -A, so (2) can be written as A + (-A) = 0 The matrix -A is called the negative of A. we also note that -A is (-1) A. Proof (a) Let A = [a¡j], B = [bij], A + B = C = [C₁j], and B + A = D = [dij] We must show that Cij = dij for all i, j. Now Ci¡j = aij + bij and d¡¡ = bij + a¡¡ for all I, j. Since a;; and bij are real numbers, we have aj + bij = bij + aij, which implies that Cij = d₁j; for all i, j.
Transcribed Image Text:I. Let A, B, and C be m x n matrices 2.) For each m x n matrix, A, there is a unique m x n matrix D such that A + D = 0 Properties of Matrix Addition We shall write D as -A, so (2) can be written as A + (-A) = 0 The matrix -A is called the negative of A. we also note that -A is (-1) A. Proof (a) Let A = [a¡j], B = [bij], A + B = C = [C₁j], and B + A = D = [dij] We must show that Cij = dij for all i, j. Now Ci¡j = aij + bij and d¡¡ = bij + a¡¡ for all I, j. Since a;; and bij are real numbers, we have aj + bij = bij + aij, which implies that Cij = d₁j; for all i, j.
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