Mathematics · JEE | DETERMINANTS

Let \( \Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{vmatrix} \) and let \( A_{ij} \) denote the cofactor of \( a_{ij} \). If \( a_{21} A_{11} + a_{22} A_{12} + a_{23} A_{13} = k \), then the value of \( k \) is:

A
\( 0 \)
B
\( -\Delta \)
C
\( 3\Delta \)
D
\( \Delta \)

View Answer & Solution

Correct answer · Option A

\( 0 \)

Explanation

If elements of any row are multiplied with cofactors of any other row, their sum is identically zero because it represents a determinant with two identical rows.

Let \( \Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{vmatrix} \) and let \( A_{ij} \) denote the cofactor of \( a_{ij} \). If \( a_{21} A_{11} + a_{22} A_{12} + a_{23} A_{13} = k \), then the value of \( k \) is:

Correct answer · Option A

Explanation

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Let \( \Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{vmatrix} \) and let \( A_{ij} \) denote the cofactor of \( a_{ij} \). If \( a_{21} A_{11} + a_{22} A_{12} + a_{23} A_{13} = k \), then the value of \( k \) is:

Correct answer · Option A

Explanation

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