Final answer:
The determinant of the matrix computed using cofactor expansion across the first row and down the second column yields the same result, which is -4.
Step-by-step explanation:
Computing the Determinant by Cofactor Expansion
To compute the determinant of a 3x3 matrix using a cofactor expansion across the first row, you take each element of the first row, multiply it by the determinant of the 2x2 matrix that remains after removing the row and column of that element, and consider the sign (positive or negative) based on its position. The formula for the determinant of a 3x3 matrix A is given as:
- det(A) = a11(A22A33 - A23A32) - a12(A21A33 - A23A31) + a13(A21A32 - A22A31)
Similarly, to do a cofactor expansion down the second column, you follow the same process for the elements of the second column.
Let's calculate using the provided matrix:
- First row expansion: det(A) = 3(1*(-2) - 2*5) - (-2)(2*(-2) - 2*1) + 4(2*5 - 1*2)
- Second column expansion: det(A) = -(-2)(2*(-2) - 1*4) + 1(3*(-2) - 4*1) + 5(3*2 - 2*4)
After calculations:
- First row expansion: det(A) = 3(-12) - (-2)(-4) + 4(8) = -36 + 8 + 32
- Second column expansion: det(A) = -(4) + 1(-6) + 5(2) = -4 - 6 + 10
The final determinant value for both methods should be identical.
Thus, the determinant of the given matrix using both methods is -4.