Question

If \( a_{ij} \) and \( A_{ij} \) represent the \((i,j)^\text{th}\) element and its cofactor of the matrix: \[ \begin{bmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{bmatrix}, \] respectively, then the value of \( a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23} \) is:

• A. 0
• B. -28
• C. 114
• D. -114
• E.

Hint:

The sum of the product of elements from one row and the cofactors from another row of a matrix is always zero. This is a fundamental property of determinants.

Answer 1

The Correct Option is A

The cofactors of a matrix are calculated from the determinant of submatrices formed by removing the corresponding row and column. However, we need to compute the value: \[ a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23}. \] Notice that this expression involves elements and cofactors from the first row of the matrix. The sum of the product of elements from one row and the cofactors from another row of the same matrix is always zero. This is a property of determinants, which states: \[ \sum_{j=1}^n a_{ij}A_{kj} = 0 \quad \text{for } i \neq k. \] Here, the first row elements (\(a_{11}, a_{12}, a_{13}\)) are multiplied by the second row cofactors (\(A_{21}, A_{22}, A_{23}\)). Since \( i = 1 \) and \( k = 2 \), the result is: \[ a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23} = 0. \] Hence, the correct answer is (A) 0.