If \(A=\begin{bmatrix}152&105&3\\149&25&35\\2&1&0\end{bmatrix}\). If \(A_{ij}\) denotes the co-factor of an element \(a_{ij}\) of the matrix A, then the value of \(a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23}\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 0
The expression multiplies the elements of row 1 by the cofactors of row 2. Since \(A_{21}, A_{22}, A_{23}\) do not involve row 2, the sum is the expansion along row 2 of the determinant obtained by replacing row 2 by row 1:
\( a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23} = \begin{vmatrix}152&105&3\\152&105&3\\2&1&0\end{vmatrix} \)
A determinant with two identical rows is 0. So the value is 0, whatever the entries of A.
NCERT (2026-27 edition):
Minors and Cofactors
