sympy import Matrix

Defining matrices A and B as given in the question

A = Matrix([[2, 1], [1, 0]]) B = Matrix([[-1, 2], [2, -4]])

Calculating matrix C = A * B

C = A * B

Calculating determinant of matrix C

det_C = C.det() det_CThe determinant of matrix $C$, which is the product of matrices $A$ and $B$, is $\text{det}(C) = 0$.

Let me know if you'd like further clarification or have any related questions!

Here are some questions to expand your understanding:

1. What are the properties of determinants when multiplying matrices?
2. Can a matrix with a determinant of 0 be invertible? Why or why not?
3. How would the determinant of $C$ change if we altered one element in matrix $A$ or $B$?
4. What does a determinant of 0 imply geometrically about matrix $C$?
5. How can the determinant of $C$ be verified using cofactor expansion?

Tip: Remember that the determinant of a product of matrices $A$ and $B$ is the product of their determinants: $\text{det}(AB) = \text{det}(A) \cdot \text{det}(B)$.