Question

If A = \(\begin{bmatrix} 2 & 1 \\[0.3em] 1 & 4 \\[0.3em] -1 & 6 \\[0.3em] 0&7 \end{bmatrix}\) and B= \(\begin{bmatrix} 1 & 2&3&0 \\[0.3em] 2 & -1 & 6&7\end{bmatrix}\) then

• A. only AB is defined
• B. only BA is defined.
• C. AB and BA both are defined.
• D. Neither AB nor BA is defined.

Answer 1

The Correct Option is C

To determine whether the matrix products \(AB\) and \(BA\) are defined, we need to consider the dimensions of matrices \(A\) and \(B\).

Matrix \(A\) is a \(4 \times 2\) matrix given by:

\(\begin{bmatrix} 2 & 1 \\[0.3em] 1 & 4 \\[0.3em] -1 & 6 \\[0.3em] 0&7 \end{bmatrix}\)

Matrix \(B\) is a \(2 \times 4\) matrix given by:

\(\begin{bmatrix} 1 & 2&3&0 \\[0.3em] 2 & -1 & 6&7\end{bmatrix}\)

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix.

Checking \(AB\):

The number of columns in \(A\) (2 columns) is equal to the number of rows in \(B\) (2 rows). Therefore, the product \(AB\) is defined. The resulting matrix will have dimensions \(4 \times 4\) (rows of \(A\) by columns of \(B\)).

Checking \(BA\):

The number of columns in \(B\) (4 columns) does not equal the number of rows in \(A\) (4 rows). Therefore, the product \(BA\) is also defined. The resulting matrix will have dimensions \(2 \times 2\) (rows of \(B\) by columns of \(A\)).

Thus, both \(AB\) and \(BA\) are defined.

The correct answer is: AB and BA both are defined.