Let's work through the problems one by one:

Exercise 2.4.9: In each case, either prove the assertion or give an example showing that it is false.

a. If $A \neq 0$ is a square matrix, then $A$ is invertible.

False. A square matrix being non-zero does not guarantee invertibility. For example, consider the matrix: $A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ This is a square matrix, and it is not the zero matrix, but it is not invertible because its determinant is 0 (a requirement for invertibility is that the determinant must be non-zero).

b. If $A$ and $B$ are both invertible, then $A + B$ is invertible.

False. The sum of two invertible matrices is not always invertible. Consider the following counterexample: $A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$ Both $A$ and $B$ are invertible, but: $A + B = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$ The resulting matrix is the zero matrix, which is not invertible.

c. If $A$ and $B$ are both invertible, then $(A^{-1}B)^T$ is invertible.

True. If both $A$ and $B$ are invertible, then their product is also invertible. We know that: $(A^{-1}B)^T = B^T (A^{-1})^T$ Both $B^T$ and $(A^{-1})^T$ are invertible since the transpose of an invertible matrix is also invertible. Therefore, the transpose $(A^{-1}B)^T$ is invertible.

d. If $A^4 = 3I$, then $A$ is invertible.

True. If $A^4 = 3I$, then: $A^4 = 3I \implies \frac{1}{3} A^4 = I$ So, $A^4$ is a scalar multiple of the identity matrix, meaning that $A$ must be invertible (since we can multiply by $\frac{1}{\sqrt[4]{3}}$ and still have a valid inverse matrix).

e. If $A^2 = A$ and $A \neq 0$, then $A$ is invertible.

False. This statement is generally false. Consider the matrix: $A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ Clearly: $A^2 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} = A$ and $A \neq 0$. However, $A$ is not invertible because its determinant is 0.

Do you want more details or have any further questions?

Here are five related questions to expand your understanding:

1. What are the conditions for a matrix to be invertible?
2. Can the sum of more than two invertible matrices be invertible?
3. How does the property of the transpose affect invertibility?
4. Can a matrix with non-zero determinant always be invertible?
5. How does the rank of a matrix relate to its invertibility?

Tip: Always check the determinant of a matrix when determining if it is invertible!