Question

If \( A = \begin{bmatrix} 1 & 2 & -1 \\ 3 & 4 & 2 \\ 2 & 0 & 1 \end{bmatrix} \) and \( B = \begin{bmatrix} 1 & 3 \\ -4 & 0 \\ 2 & 5 \end{bmatrix} \) are two matrices, then which one of the following is incorrect:

• A. AB is defined
• B. BA is not defined
• C. A + B is not defined
• D. A - B is defined

Hint:

- Matrix multiplication \(M_{m \times n} \times N_{p \times q}\) is defined only if \(n=p\). - Matrix addition/subtraction is defined only if the matrices have the same dimensions.

Answer 1

The Correct Option is D

Step 1: Determine the dimensions of matrices A and B.
Matrix A is a \(3 \times 3\) matrix (3 rows, 3 columns).
Matrix B is a \(3 \times 2\) matrix (3 rows, 2 columns).

Step 2: Verify the validity of each operation.
(A) AB is defined because the number of columns in A (3) equals the number of rows in B (3). The resulting matrix has dimensions \(3 \times 2\). This statement is correct.
(B) BA is not defined because the number of columns in B (2) does not equal the number of rows in A (3). This statement is correct.
(C) A + B is not defined because matrices A (\(3 \times 3\)) and B (\(3 \times 2\)) have different dimensions. This statement is correct.
(D) A - B is not defined because matrices A (\(3 \times 3\)) and B (\(3 \times 2\)) have different dimensions. The statement incorrectly claims A - B is defined.