If C_ij represents the cofactor of element a_ij of the matrix A = bmatrix 2 & -1 & 3 \\ 1 & 2 & 0 \\ 4 & 1 & 5 bmatrix then the value of C_23 + C_31 - C_22 is Option 1: 14 Option 2: -4 Option 3: -10 Option 4: 8

Answer: -10. Solution: The cofactor C_ij is given by (-1)^(i+j) M_ij, where M_ij is the determinant of the submatrix after removing the i-th row and j-th column. The matrix is: A = bmatrix 2 & -1 & 3 \\ 1 & 2 & 0 \\ 4 & 1 & 5 bmatrix C_23 = (-1)^(2+3) vmatrix 2 & -1 \\ 4 & 1 vmatrix C_23 = -1(2(1) - (-1)(4)) C_23 = -1(2 + 4) C_23 = -6 C_31 = (-1)^(3+1) vmatrix -1 & 3 \\ 2 & 0 vmatrix C_31 = 1((-1)(0) - (3)(2)) C_31 = 1(0 - 6) C_31 = -6 C_22 = (-1)^(2+2) vmatrix 2 & 3 \\ 4 & 5 vmatrix C_22 = 1(2(5) - (3)(4)) C_22 = 1(10 - 12) C_22 = -2 C_23 + C_31 - C_22 = -6 + (-6) - (-2) = -6 - 6 + 2 = -10

Algebra

Matrices & Determinants

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If $C_{ij}$ represents the cofactor of element $a_{ij}$ of the matrix $A = \begin{bmatrix} 2 & -1 & 3 \\ 1 & 2 & 0 \\ 4 & 1 & 5 \end{bmatrix}$ then the value of $C_{23} + C_{31} - C_{22}$ is

-4

-10

✅ Correct Option: 3

The cofactor $C_{ij}$ is given by $(-1)^{i+j} M_{ij}$ , where $M_{ij}$ is the determinant of the submatrix after removing the $i$ -th row and $j$ -th column.

The matrix is:

$A = \begin{bmatrix} 2 & -1 & 3 \\ 1 & 2 & 0 \\ 4 & 1 & 5 \end{bmatrix}$

$C_{23} = (-1)^{2+3} \begin{vmatrix} 2 & -1 \\ 4 & 1 \end{vmatrix}$

$C_{23} = -1(2(1) - (-1)(4))$

$C_{23} = -1(2 + 4)$

$C_{23} = -6$

$C_{31} = (-1)^{3+1} \begin{vmatrix} -1 & 3 \\ 2 & 0 \end{vmatrix}$

$C_{31} = 1((-1)(0) - (3)(2))$

$C_{31} = 1(0 - 6)$

$C_{31} = -6$

$C_{22} = (-1)^{2+2} \begin{vmatrix} 2 & 3 \\ 4 & 5 \end{vmatrix}$

$C_{22} = 1(2(5) - (3)(4))$

$C_{22} = 1(10 - 12)$

$C_{22} = -2$

$C_{23} + C_{31} - C_{22} = -6 + (-6) - (-2)$

$= -6 - 6 + 2$

$= -10$

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If $C_{ij}$ represents the cofactor of element $a_{ij}$ of the matrix $A = \begin{bmatrix} 2 & -1 & 3 \\ 1 & 2 & 0 \\ 4 & 1 & 5 \end{bmatrix}$ then the value of $C_{23} + C_{31} - C_{22}$ is