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To solve for \( X \), \( Y \), and \( Z \) in terms of \( A \), \( B \), and \( C \), let's analyze the matrix equation provided. The equation is:

\[
\begin{bmatrix} X & 0 \\ Y & Z \end{bmatrix}
\begin{bmatrix} A & 0 \\ B & C \end{bmatrix} =
\begin{bmatrix} I & 0 \\ 0 & I \end{bmatrix}
\]

### Step 1: Matrix multiplication

We multiply the two matrices on the left-hand side:

\[
\begin{bmatrix} X & 0 \\ Y & Z \end{bmatrix}
\begin{bmatrix} A & 0 \\ B & C \end{bmatrix}
=
\begin{bmatrix} X A & 0 \\ Y A + Z B & Z C \end{bmatrix}
\]

### Step 2: Set equal to the identity matrix

We now set this product equal to the right-hand side, which is the identity matrix:

\[
\begin{bmatrix} X A & 0 \\ Y A + Z B & Z C \end{bmatrix}
=
\begin{bmatrix} I & 0 \\ 0 & I \end{bmatrix}
\]

This gives us two systems of equations by comparing the corresponding entries:

1. \( X A = I \)
2. \( Y A + Z B = 0 \)
3. \( Z C = I \)

### Step 3: Solve the equations

1. From \( X A = I \), we solve for \( X \):
   \[
   X = A^{-1}
   \]
   assuming that \( A \) is invertible.

2. From \( Z C = I \), we solve for \( Z \):
   \[
   Z = C^{-1}
   \]
   assuming that \( C \) is invertible.

3. Substitute \( X = A^{-1} \) and \( Z = C^{-1} \) into \( Y A + Z B = 0 \):
   \[
   Y A + C^{-1} B = 0
   \]
   Solve for \( Y \):
   \[
   Y A = -C^{-1} B
   \]
   \[
   Y = -C^{-1} B A^{-1}
   \]

### Final Answer:

- \( X = A^{-1} \)
- \( Y = -C^{-1} B A^{-1} \)
- \( Z = C^{-1} \)

Would you like more details or have any questions? Here are some follow-up questions to consider:

1. What happens if \( A \) or \( C \) are not invertible?
2. How would the solution change if \( B = 0 \)?
3. Can you find the inverse of a specific matrix \( A \)?
4. How does matrix multiplication order affect the result?
5. What conditions must be met for these matrices to be invertible?

**Tip:** Remember that matrix inversion only exists if the matrix is square and non-singular (i.e., it has full rank).

Find formulas for X, Y, and Z in terms of A, B, and C given the matrix equation: [X 0; Y Z] [A 0; B C] = [I 0; 0 I].

Solve the matrix equation [X 0; Y Z] [A 0; B C] = [I 0; 0 I] and find formulas for X, Y, and Z in terms of matrices A, B, and C. Learn key concepts from linear algebra such as matrix multiplication and inverses. Suitable for university-level students.

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