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.