Find formulas for $X, Y$, and $Z$ in terms of $A, B$, and $C$. It may be necessary to make assumptions about the size of a matrix in order to produce a formula. [Hint Compute the product on the left, and set it equal to the right side.] \[ \left[\begin{array}{cc} A & B \\ C & 0 \end{array}\right]\left[\begin{array}{cc} I & 0 \\ X & Y \end{array}\right]=\left[\begin{array}{ll} 0 & I \\ Z & 0 \end{array}\right] \] Find the formulas for $X, Y$, and $Z$. Note that I represents the identity matrix and 0 represents the zero matrix. \[ \begin{array}{l} X= \\ Y= \\ Z= \end{array} \]

Step 1 ：We are given the matrix equation \(\left[\begin{array}{cc} A & B \\ C & 0 \end{array}\right]\left[\begin{array}{cc} I & 0 \\ X & Y \end{array}\right] = \left[\begin{array}{ll} 0 & I \\ Z & 0 \end{array}\right]\).

Step 2 ：We multiply the matrices on the left side to get \(\left[\begin{array}{cc} AI + BX & BY \\ CI & 0 \end{array}\right]\).

Step 3 ：We set this equal to the matrix on the right side of the equation to get \(\left[\begin{array}{cc} AI + BX & BY \\ CI & 0 \end{array}\right] = \left[\begin{array}{ll} 0 & I \\ Z & 0 \end{array}\right]\).

Step 4 ：From this, we can form the following equations: \(AI + BX = 0\), \(BY = I\), \(CI = Z\), and \(0 = 0\).

Step 5 ：Solving these equations, we find that \(X = -\frac{A \cdot I}{B}\), \(Y = \frac{I}{B}\), and \(Z = C \cdot I\).

Step 6 ：Thus, the solutions for X, Y, and Z are \(\boxed{X = -\frac{A \cdot I}{B}}\), \(\boxed{Y = \frac{I}{B}}\), and \(\boxed{Z = C \cdot I}\).