If $c_{ij}$ denotes the cofactor of element $a_{ij}$ of the matrix $A =\begin{bmatrix}1&2&-1\\0&-3&2\\4&2&3\end{bmatrix}$, then the value of $c_{21}.c_{33}$ is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 24
Matrix $A = \begin{bmatrix} 1 & 2 & -1 \\ 0 & -3 & 2 \\ 4 & 2 & 3 \end{bmatrix}$
$c_{21}$ is the cofactor of $a_{21} = 0$
Minor = $\begin{vmatrix} 2 & -1 \\ 2 & 3 \end{vmatrix} = (2)(3) - (-1)(2) = 6 + 2 = 8$
$c_{21} = (-1)^{2+1} \cdot 8 = -8$
$c_{33}$ is the cofactor of $a_{33} = 3$
Minor = $\begin{vmatrix} 1 & 2 \\ 0 & -3 \end{vmatrix} = (1)(-3) - (2)(0) = -3$
$c_{33} = (-1)^{3+3} \cdot (-3) = 1 \cdot (-3) = -3$
So, $c_{21} \cdot c_{33} = (-8)(-3) = \mathbf{24}$