Question

If \(A = [a_{ij}] = \begin{bmatrix} 2 & -1 & 5 \\ 1 & 3 & 2 \\ 5 & 0 & 4 \end{bmatrix}\) and \( c_{ij} \) is the cofactor of element \( a_{ij} \), then the value of \( a_{21}c_{11} + a_{22}c_{12} + a_{23}c_{13} \) is:

• A. \( -57 \)
• B. \( 0 \)
• C. \( 9 \)
• D. \( 57 \)

Hint:

For determinant calculations, remember that the sum of the product of the elements of any row or column with the cofactors of another row or column is always zero.

Answer 1

The Correct Option is B

Step 1: Understanding the expression. The given expression represents the sum of products of elements of the second row with the cofactors of the corresponding elements from the first row. 

Step 2: Determinant Property. By the cofactor expansion property: \[ a_{21}c_{11} + a_{22}c_{12} + a_{23}c_{13} = 0 \] since this is equivalent to the determinant expansion along a different row of the same matrix. 

Conclusion: Thus, the required value is \( 0 \), which corresponds to option \( \mathbf{(B)} \).