I will solve the problems one by one as stated in the image. Let me begin by calculating the determinants of $A_1$ and $A_2$. numpy as np

Define the matrices A1 and A2

A1 = np.array([[2, 0, 0], [1, 2, -1], [1, 3, -2]]) A2 = np.array([[3, 12, -21], [-1, -6, 13], [0, -2, 6]])

Calculate the determinants of A1 and A2

det_A1 = np.linalg.det(A1) det_A2 = np.linalg.det(A2)

det_A1, det_A2The determinants of the matrices are as follows:

1. $\det(A_1) = -2$
2. $\det(A_2) = 0$

This shows that $A_1$ is invertible, while $A_2$ is not.

Would you like to proceed to the next question (characteristic polynomials) or review the above? Here are some related questions:

1. Why does a determinant of zero imply a matrix is not invertible?
2. What is the role of determinants in solving systems of equations?
3. How does the determinant relate to eigenvalues?
4. Can we deduce properties of eigenvectors from the determinant alone?
5. How do determinants scale when a matrix is multiplied by a scalar?

Tip: The determinant provides insight into whether a matrix transformation is singular or non-singular.