The problem asks to find a matrix \( C \) such that:

\[
(B + C)^{-1} = A
\]

### Step 1: Start by expressing the equation in terms of \( B + C \).
Taking the inverse of both sides, we get:

\[
B + C = A^{-1}
\]

### Step 2: Use the previously calculated inverse of \( A \).
From part (a), we found:

\[
A^{-1} = \begin{pmatrix} 1 & -1 \\ -3 & 4 \end{pmatrix}
\]

### Step 3: Express \( C \) in terms of \( B \) and \( A^{-1} \).
We can rearrange the equation to solve for \( C \):

\[
C = A^{-1} - B
\]

### Step 4: Substitute the given matrices \( B \) and \( A^{-1} \) into the equation.
Given:

\[
B = \begin{pmatrix} 2 & -1 \\ -3 & 5 \end{pmatrix}
\]
and
\[
A^{-1} = \begin{pmatrix} 1 & -1 \\ -3 & 4 \end{pmatrix}
\]

We compute \( C \) as follows:

\[
C = \begin{pmatrix} 1 & -1 \\ -3 & 4 \end{pmatrix} - \begin{pmatrix} 2 & -1 \\ -3 & 5 \end{pmatrix}
\]

### Step 5: Perform the matrix subtraction.
Subtract the corresponding elements of \( B \) from \( A^{-1} \):

\[
C = \begin{pmatrix} 1 - 2 & -1 - (-1) \\ -3 - (-3) & 4 - 5 \end{pmatrix}
\]

\[
C = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}
\]

### Final Answer:
The matrix \( C \) that satisfies the equation \( (B + C)^{-1} = A \) is:

\[
C = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}
\]

Would you like more details or have any questions? Here are some related questions that might interest you:

1. How would you find \( C \) if the matrix \( B \) were different?
2. What properties must matrix \( C \) have to satisfy the equation \( (B + C)^{-1} = A \)?
3. Can you find \( B + C \) and verify that its inverse is indeed \( A \)?
4. What is the significance of finding such a matrix \( C \) in linear algebra?
5. How would the solution change if \( A \) were not invertible?

**Tip:** When subtracting matrices, ensure that corresponding elements are subtracted from one another, and the result is another matrix of the same dimensions.

Learn how to find matrix C such that (B+C)^{-1} = A with a detailed step-by-step solution. This problem involves matrix algebra and the calculation of matrix inverses. Suitable for college-level students and those interested in linear algebra.

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