If $C_{ij}$ represents the cofactor of element $a_{ij}$ of the matrix $A =\begin{bmatrix}2&-1&3\\1&2&0\\4&1&5\end{bmatrix}$ then the value of $C_{23}+ C_{31}-C_{22}$ is

-10

The correct answer is Option (3) → -10

Given:

$A=\begin{bmatrix} 2 & -1 & 3 \\ 1 & 2 & 0 \\ 4 & 1 & 5 \end{bmatrix}$

$C_{23}$ = cofactor of $a_{23}$ = $0$ in row 2, column 3 → minor from $\begin{bmatrix} 2 & -1 \\ 4 & 1 \end{bmatrix}$

$M_{23} = (2)(1) - (-1)(4) = 2 + 4 = 6$

Sign factor = $(-1)^{2+3} = -1$

$C_{23} = -6$

$C_{31}$ = cofactor of $a_{31}$ = $4$ in row 3, column 1 → minor from $\begin{bmatrix} -1 & 3 \\ 2 & 0 \end{bmatrix}$

$M_{31} = (-1)(0) - (3)(2) = 0 - 6 = -6$

Sign factor = $(-1)^{3+1} = 1$

$C_{31} = -6$

$C_{22}$ = cofactor of $a_{22}$ = $2$ in row 2, column 2 → minor from $\begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}$

$M_{22} = (2)(5) - (3)(4) = 10 - 12 = -2$

Sign factor = $(-1)^{2+2} = 1$

$C_{22} = -2$

Required value: $C_{23} + C_{31} - C_{22} = (-6) + (-6) - (-2) = -6 - 6 + 2 = -10$