Question-32

Let \(A=\begin{bmatrix} 1 & 2 & 3 & 4\\ 4 & 1 & 2 & 3\\ 3 & 4 & 1 & 2\\ 2 & 3 & 4 & 1 \end{bmatrix}\) and \(B=\begin{bmatrix} 3 & 4 & 1 & 2\\ 4 & 1 & 2 & 3\\ 1 & 2 & 3 & 4\\ 2 & 3 & 4 & 1 \end{bmatrix}\). If \(\text{det}( A)\) and \(\text{det}( B)\) denotes the determinants of these two matrices, select the true statement from the options given below.

• \(\text{det}( A) =\text{det}( B)\)

• \(\text{det}( A) =-\text{det}( B)\)

• \(\text{det}( A) =0\)

• \(\text{det}( AB) =\text{det}( A) +\text{det}( B)\)

Answer

• \(\text{det}( A) =\text{det}( B)\)

• \(\text{det}( A) =-\text{det}( B)\)

• \(\text{det}( A) =0\)

• \(\text{det}( AB) =\text{det}( A) +\text{det}( B)\)

Solution

\(A\) and \(B\) have the order of rows \(1\) and \(3\) swapped. Swapping rows changes the sign of the determinant.