Final answer:
To compute the determinant of the matrix A using the recursive definition, you can use the cofactor expansion. The determinant of matrix A is 0.
Step-by-step explanation:
To compute the determinant of the matrix A = [[3, -1, 8], [-4, 6, 0], [6, -9, 0]] using the recursive definition, you can use the cofactor expansion.
First, let's expand along the first row:
det(A) = 3 * det([[6, 0], [-9, 0]]) - (-1) * det([[-4, 0], [6, 0]]) + 8 * det([[-4, 6], [6, -9]])
Next, we can compute the determinants of the 2x2 matrices using the formula ad - bc:
det([[6, 0], [-9, 0]]) = 6 * 0 - (-9) * 0 = 0
det([[-4, 0], [6, 0]]) = -4 * 0 - 0 * 6 = 0
det([[-4, 6], [6, -9]]) = -4 * (-9) - 6 * 6 = -36 + 36 = 0
Therefore, the determinant of matrix A is:
det(A) = 3 * 0 - (-1) * 0 + 8 * 0 = 0