Determinant of 4x4 Matrix Calculation

To find the determinant of the given 4x4 matrix, we use the cofactor expansion method. The matrix is: $ A = \begin{pmatrix} 1 & 0 & -1 & 0 \\ 4 & 7 & 0 & 2 \\ 1 & 1 & -1 & 1 \\ 2 & 0 & 2 & 1 \end{pmatrix} $ Expanding along the first row ($R_1$) is efficient due to the presence of zeros.

Cofactor Expansion along First Row

The determinant formula using cofactor expansion along the first row is: $ \det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} + a_{14}C_{14} $ Where $ a_{ij} $ is the element in the i-th row and j-th column, and $ C_{ij} = (-1)^{i+j} M_{ij} $ is the cofactor, with $ M_{ij} $ being the minor. Since $ a_{12}=0 $ and $ a_{14}=0 $, the formula simplifies to: $ \det(A) = a_{11}C_{11} + a_{13}C_{13} $ $ \det(A) = (1)(-1)^{1+1} M_{11} + (-1)(-1)^{1+3} M_{13} $ $ \det(A) = M_{11} - M_{13} $

Calculating the Minors

1. Minor $ M_{11} $: This is the determinant of the submatrix formed by removing the first row and first column of A. $ M_{11} = \| \begin{vmatrix} 7 & 0 & 2 \\ 1 & -1 & 1 \\ 0 & 2 & 1 \end{vmatrix} \| $ Calculating this 3x3 determinant: $ M_{11} = 7 \times \| \begin{vmatrix} -1 & 1 \\ 2 & 1 \end{vmatrix} \| - 0 + 2 \times \| \begin{vmatrix} 1 & -1 \\ 0 & 2 \end{vmatrix} \| $ $ M_{11} = 7 ((-1)(1) - (1)(2)) + 2 ((1)(2) - (-1)(0)) $ $ M_{11} = 7 (-3) + 2 (2) = -21 + 4 = -17 $

2. Minor $ M_{13} $: This is the determinant of the submatrix formed by removing the first row and third column of A. $ M_{13} = \| \begin{vmatrix} 4 & 7 & 2 \\ 1 & 1 & 1 \\ 2 & 0 & 1 \end{vmatrix} \| $ Calculating this 3x3 determinant: $ M_{13} = 2 \times \| \begin{vmatrix} 7 & 2 \\ 1 & 1 \end{vmatrix} \| - 0 + 1 \times \| \begin{vmatrix} 4 & 7 \\ 1 & 1 \end{vmatrix} \| $ $ M_{13} = 2 ((7)(1) - (2)(1)) + 1 ((4)(1) - (7)(1)) $ $ M_{13} = 2 (5) + 1 (-3) = 10 - 3 = 7 $

Determining the Determinant Value

Now, substitute the calculated minors back into the simplified formula: $ \det(A) = M_{11} - M_{13} $ $ \det(A) = -17 - 7 = -24 $ The value obtained from the calculation is -24. The correct option is indicated as D.