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

Cofactors of a matrix are derived from the determinants of submatrices obtained by removing the corresponding row and column. The expression to compute is: \[a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23}.\] This expression combines elements from the first row of the matrix with cofactors from the second row. A property of determinants states that the sum of the products of elements from one row and the cofactors from a different row of the same matrix is always zero: \[\sum_{j=1}^n a_{ij}A_{kj} = 0 \quad \text{for } i eq k.\] In this case, elements from the first row (\(a_{11}, a_{12}, a_{13}\)) are multiplied by cofactors from the second row (\(A_{21}, A_{22}, A_{23}\)). Since \(i = 1\) and \(k = 2\), satisfying \(i eq k\), the result is: \[a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23} = 0.\] Therefore, the correct answer is (A) 0.