Question

If \[ A = \begin{bmatrix} 1 & 2 & 3 \\ -4 & 3 & 7 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & 3 \\ -1 & 2 \\ 0 & 5 \end{bmatrix}, \] then the correct statement is:

• A. Only AB is defined.
• B. Only BA is defined.
• C. AB and BA, both are defined.
• D. AB and BA, both are not defined.

Hint:

Always check the dimensions of matrices before attempting to multiply them. The number of columns in the first matrix must equal the number of rows in the second matrix for multiplication to be possible.

Answer 1

The Correct Option is A

To determine whether the matrix products \( AB \) and \( BA \) are defined, we need to check the dimensions of the matrices.
- Matrix \( A \) has dimensions \( 2 \times 3 \) (2 rows and 3 columns). 
- Matrix \( B \) has dimensions \( 3 \times 2 \) (3 rows and 2 columns). 
For the product \( AB \) to be defined, the number of columns of \( A \) must match the number of rows of \( B \). In this case, \( A \) has 3 columns and \( B \) has 3 rows, so \( AB \) is defined. 
The resulting matrix will have dimensions \( 2 \times 2 \). For the product \( BA \), the number of columns of \( B \) must match the number of rows of \( A \). 
However, \( B \) has 2 columns and \( A \) has 2 rows, so \( BA \) is not defined. Hence, only \( AB \) is defined.