Question

In the determinant \[ \begin{vmatrix} 3 & x & -1 2 & -1 & 4 1 & y & -3 \end{vmatrix} \] the sum of the cofactors of \(x\) and \(y\) is

• A. -24
• B. 24
• C. -4
• D. 4

Hint:

Remember checkerboard sign pattern for cofactors.

Answer 1

The Correct Option is C

Concept: Cofactor of element.

Step 1: Cofactor of \(x\).
\[ \begin{vmatrix} 2 & 4 \\ 1 & -3 \end{vmatrix} = (2)(-3) - (4)(1) = -6 - 4 = -10 \] Sign: \[ C_{12} = +10 \]

Step 2: Cofactor of \(y\).
\[ \begin{vmatrix} 3 & -1 \\ 2 & 4 \end{vmatrix} = 12 + 2 = 14 \] Sign: \[ C_{32} = -14 \]

Step 3: Sum.
\[ 10 - 14 = -4 \]