Question

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problem 6. (12 points)
let ( a = \begin{bmatrix} 2 & 7 \\ -3 & -7 \\ -2 & -8 end{bmatrix} ) and ( b = \begin{bmatrix} 1 & -5 \\ -1 & -1 end{bmatrix} ).
find ( a - b ) if possible. if it is not possible to perform this operation, explain.
select the correct choice below, and, if necessary, fill in the answer boxes to complete your choice.
( \bigcirc ) a. ( a - b = \begin{bmatrix} square & square \\ square & square \\ square & square end{bmatrix} )
( \bigcirc ) b. the operation is not defined because the number of columns in the first matrix is not the same as the number of rows in the second matrix.
( \bigcirc ) c. the difference of two matrices of different sizes is not defined.
( \bigcirc ) d. the operation is not defined because the number of rows in the first matrix is not the same as the number of columns in the second matrix.
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Explanation:

Step1: Check Matrix Sizes

Matrix \( A \) has dimensions \( 3 \times 2 \) (3 rows, 2 columns), matrix \( B \) has dimensions \( 2 \times 2 \) (2 rows, 2 columns). For matrix subtraction \( A - B \), matrices must have the same dimensions (same number of rows and columns). Since \( A \) is \( 3 \times 2 \) and \( B \) is \( 2 \times 2 \), their sizes differ.

Step2: Determine Validity

Matrix subtraction (and addition) is only defined for matrices of the same size. Since \( A \) and \( B \) have different numbers of rows (3 vs. 2), the operation \( A - B \) is not defined because the difference of two matrices of different sizes is not defined.

Answer:

C. The difference of two matrices of different sizes is not defined.