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Let's break down the problem from the image step by step.

### Given Information:
You are working with \( n \times n \) matrices \( A \), \( B \), and \( X \). It is given that:
- \( A - AX \) is invertible.
- The equation is: 
  \[
  (A - AX)^{-1} = X^{-1} B \tag{3}
  \]

The problem asks:

**(a)** Explain why \( B \) is invertible.

**(b)** Solve equation (3) for \( X \). If you need to invert a matrix in the process, explain why that matrix is invertible.

### Part (a): Why is \( B \) invertible?

To explain why \( B \) is invertible, we start with the given equation:
\[
(A - AX)^{-1} = X^{-1}B
\]
We know that the left-hand side of this equation involves an inverse, meaning that \( A - AX \) must be invertible by assumption. The right-hand side involves \( X^{-1} \), meaning that \( X \) is also invertible, which leads us to infer that \( B \) must be invertible in order to maintain the equality of matrix dimensions and inverses. 

Thus, since \( X^{-1} B \) exists and we are assuming \( X \) is invertible, \( B \) must also be invertible.

### Part (b): Solving for \( X \)

To solve for \( X \), let's start with the equation:
\[
(A - AX)^{-1} = X^{-1}B
\]

1. **Multiply both sides by \( (A - AX) \) on the left**:
   \[
   I = (A - AX)X^{-1}B
   \]
   This simplifies to:
   \[
   I = (A - AX)X^{-1}B
   \]
   
2. **Now multiply both sides by \( B^{-1} \)** (since \( B \) is invertible, as we've shown):
   \[
   B^{-1} = (A - AX)X^{-1}
   \]
   
3. **Next, multiply both sides by \( X \) from the right**:
   \[
   B^{-1}X = A - AX
   \]

4. **Now, solve for \( X \)**. Collect the \( X \)-terms on one side:
   \[
   B^{-1}X + AX = A
   \]
   Factor out \( X \) on the left:
   \[
   (B^{-1} + A)X = A
   \]
   
5. **Finally, solve for \( X \)** by multiplying both sides by \( (B^{-1} + A)^{-1} \), assuming \( (B^{-1} + A) \) is invertible:
   \[
   X = (B^{-1} + A)^{-1} A
   \]

### Why is \( (B^{-1} + A) \) invertible?

We assume that \( A - AX \) is invertible, and since \( X = (B^{-1} + A)^{-1}A \), this ensures that the necessary matrix for inversion \( (B^{-1} + A) \) can be considered invertible under the given conditions of the problem.

---

Would you like more details on any step? Also, here are five related questions you might find helpful:

1. How can you verify if a matrix is invertible in general?
2. Can you compute an example with specific \( A \), \( B \), and \( X \) matrices?
3. What conditions are necessary for a matrix to have an inverse?
4. Can you explore if \( B^{-1} + A \) is always invertible under certain conditions?
5. How do you interpret matrix inversions in terms of linear transformations?

**Tip:** When solving matrix equations, always check if the matrices involved are invertible before performing any operations involving inverses.

Suppose A, B, and X are n x n matrices with A, X, and A - AX invertible, and (A - AX)^-1 = X^-1 B. Explain why B is invertible and solve for X.

Explore how to prove the invertibility of matrix B and solve for matrix X in the equation (A - AX)^-1 = X^-1 B. This solution involves matrix algebra and uses the invertible matrix theorem. Suitable for undergraduate students.

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