Questions: Find the determinant of each matrix A=[[-2, 0, 0, 0], [2, -4, 0, 0], [-4, -8, 1, 0], [2, -2, -9, -3]] then det(B)=

Transcript text: Find the determinant of each matrix \[ \begin{array}{l} A=\left[\begin{array}{cccc} -2 & 0 & 0 & 0 \\ 2 & -4 & 0 & 0 \\ -4 & -8 & 1 & 0 \\ 2 & -2 & -9 & -3 \end{array}\right] \end{array} \] \[ \text { then } \operatorname{det}(B)= \]

Solution

Solution Steps

To find the determinant of each matrix, we can use a Python library such as NumPy, which has built-in functions to compute the determinant of a matrix. We will define each matrix and then use the numpy.linalg.det function to calculate their determinants.

Step 1: Determinant of Matrix A

To find the determinant of matrix \( A \): \[ A = \begin{bmatrix} -2 & 0 & 0 & 0 \\ 2 & -4 & 0 & 0 \\ -4 & -8 & 1 & 0 \\ 2 & -2 & -9 & -3 \end{bmatrix} \] The calculated determinant is: \[ \operatorname{det}(A) \approx -24.0 \]

Step 2: Determinant of Matrix B

Next, we calculate the determinant of matrix \( B \): \[ B = \begin{bmatrix} 4 & 0 & 0 & 0 & 0 \\ 8 & -2 & 0 & 0 & 0 \\ -9 & 9 & 6 & 0 & 0 \\ -9 & -4 & 5 & 1 & 0 \\ 1 & 7 & -8 & -7 & 2 \end{bmatrix} \] The calculated determinant is: \[ \operatorname{det}(B) \approx -96.0 \]