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
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → −10
The cofactor is \(C_{ij} = (-1)^{i+j}M_{ij}\), where the minor \(M_{ij}\) is the determinant left after deleting row i and column j.
\(\displaystyle C_{23} = (-1)^{5}\begin{vmatrix}2&-1\\4&1\end{vmatrix} = -(2 + 4) = -6\)
\(\displaystyle C_{31} = (-1)^{4}\begin{vmatrix}-1&3\\2&0\end{vmatrix} = 0 - 6 = -6\)
\(\displaystyle C_{22} = (-1)^{4}\begin{vmatrix}2&3\\4&5\end{vmatrix} = 10 - 12 = -2\)
So
\(C_{23} + C_{31} - C_{22} = -6 - 6 + 2 = -10\)