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Chapter 3: Problem 55

If \(A\) and \(B\) are \(2 \times 3\) matrices and \(A=B\), what can you say about \(A-B ?\) Explain.

Short Answer

Expert verified
If \(A\) and \(B\) are 2x3 matrices and \(A = B\), then \(A - B\) is a 2x3 matrix with all elements equal to zero. This is because the subtraction of corresponding elements in matrices \(A\) and \(B\) results in zero since their elements are equal.

Step by step solution

01

Define Matrix Subtraction

Matrix subtraction is the process of subtracting corresponding elements of two matrices. The matrices need to have the same dimensions for this operation to be defined. In this case, both A and B have dimensions 2x3. Let's name the elements of matrices A and B as follows: Matrix A: \(\begin{bmatrix} a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\end{bmatrix}\) Matrix B: \(\begin{bmatrix} b_{11}&b_{12}&b_{13}\\ b_{21}&b_{22}&b_{23}\end{bmatrix}\)
02

Deduce the Result of A - B If A = B

Given that A = B, each element in A is equal to the corresponding element in B. Therefore, we have: \(a_{11} = b_{11}, a_{12} = b_{12}, a_{13} = b_{13}\) \(a_{21} = b_{21}, a_{22} = b_{22}, a_{23} = b_{23}\) Now, let's subtract matrix B from matrix A: A - B = \(\begin{bmatrix} a_{11}-b_{11}&a_{12}-b_{12}&a_{13}-b_{13}\\ a_{21}-b_{21}&a_{22}-b_{22}&a_{23}-b_{23}\end{bmatrix}\) Since A = B, the subtraction for each element results in zero: A - B = \(\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\) Thus, if \(A = B\), then \(A - B\) is a 2x3 matrix with all elements equal to zero.

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