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

To solve this problem, we need to compute the sum of the cofactors of \( x \) and \( y \) in the given determinant.

3	x	-1
2	-1	4
1	y	-3

The determinant of a \( 3 \times 3 \) matrix is calculated as:

\(\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg)\)

In this specific problem, the matrix is:

\(\begin{vmatrix} 3 & x & -1 \\ 2 & -1 & 4 \\ 1 & y & -3 \end{vmatrix}\)

To find the cofactor of any element \( a_{ij} \), denoted by \( C_{ij} \), we omit the \( i \)-th row and \( j \)-th column containing the element and calculate the determinant of the resulting \( 2 \times 2 \) matrix, multiplied by \( (-1)^{i+j} \).

Calculate the cofactor of \( x \) (second element of the first row, \( a_{12} \)):

• The minor of \( x \) is the determinant obtained by removing row 1 and column 2:

\(\begin{vmatrix} 2 & 4 \\ 1 & -3 \end{vmatrix} = (2)(-3) - (4)(1) = -6 - 4 = -10\)

• The cofactor \( C_{12} \) of \( x \) is given by \( (-1)^{1+2} (-10) = -(-10) = 10 \).

Calculate the cofactor of \( y \) (second element of the third row, \( a_{32} \)):

• The minor of \( y \) is the determinant obtained by removing row 3 and column 2:

\(\begin{vmatrix} 3 & -1 \\ 2 & 4 \end{vmatrix} = (3)(4) - (2)(-1) = 12 + 2 = 14\)

• The cofactor \( C_{32} \) of \( y \) is given by \( (-1)^{3+2} (14) = (-1)(14) = -14 \).

The sum of cofactors of \( x \) and \( y \) is:

\(10 - 14 = -4\)

Therefore, the sum of the cofactors of \( x \) and \( y \) is -4, which corresponds to the given option, -4.