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 has 3 rows and 3 columns, so its dimension is \(3 \times 3\).
Matrix B has 3 rows and 2 columns, so its dimension is \(3 \times 2\).

Step 2: Check the condition for each operation.
(A) For the product AB to be defined, the number of columns in A must be equal to the number of rows in B. Here, A has 3 columns and B has 3 rows, so AB is defined. The resulting matrix will have dimensions \(3 \times 2\). The statement is correct.
(B) For the product BA to be defined, the number of columns in B (2) must be equal to the number of rows in A (3). Since \(2 \neq 3\), BA is not defined. The statement is correct.
(C) For matrix addition A + B to be defined, the matrices must have the same dimensions. A is \(3 \times 3\) and B is \(3 \times 2\). Since their dimensions are different, A + B is not defined. The statement is correct.
(D) For matrix subtraction A - B to be defined, the matrices must have the same dimensions. Since A is \(3 \times 3\) and B is \(3 \times 2\), A - B is not defined. The statement says that A - B is defined, which is incorrect.