https://file.aiquickdraw.com/mathbot/file/17269603942605acisuh2.jpg

Let's work through the matrix equations provided in the image step by step to find the values of \(x\), \(y\), and \(z\).

### a. 
\[
\begin{bmatrix} x & 3 \\ 1 - y & z \end{bmatrix} = \begin{bmatrix} 4 & x - 1 \\ z & -2 \end{bmatrix}
\]
Matching the elements of the two matrices:

- First row, first column: \(x = 4\)
- First row, second column: \(3 = x - 1\), so \(x = 4\) (consistent with the previous result)
- Second row, first column: \(1 - y = z\)
- Second row, second column: \(z = -2\)

From this, \(z = -2\), and substituting \(z = -2\) into \(1 - y = z\), we get:
\[
1 - y = -2 \quad \Rightarrow \quad y = 3
\]

Thus, the solution for part (a) is:
\[
x = 4, \quad y = 3, \quad z = -2
\]

---

### b.
\[
\begin{bmatrix} 9 & 2 \\ -y & z^2 \end{bmatrix} = \begin{bmatrix} x^2 & x - 1 \\ y & z \end{bmatrix}
\]
Matching the elements:

- First row, first column: \(9 = x^2\), so \(x = 3\) (taking the positive square root)
- First row, second column: \(2 = x - 1\), so \(x = 3\) (consistent with the previous result)
- Second row, first column: \(-y = y\), which implies \(y = 0\)
- Second row, second column: \(z^2 = z\)

The equation \(z^2 = z\) can be solved as:
\[
z(z - 1) = 0 \quad \Rightarrow \quad z = 0 \text{ or } z = 1
\]

Thus, the solution for part (b) is:
\[
x = 3, \quad y = 0, \quad z = 0 \text{ or } z = 1
\]

---

### c.
\[
\begin{bmatrix} x - 5 & 2 \\ 3 - y & z \end{bmatrix} = \begin{bmatrix} 2x - 11 & 2 \\ y + 9 & z \end{bmatrix}
\]
Matching the elements:

- First row, first column: \(x - 5 = 2x - 11\), solving for \(x\):
  \[
  x - 5 = 2x - 11 \quad \Rightarrow \quad -5 + 11 = x \quad \Rightarrow \quad x = 6
  \]
- First row, second column: \(2 = 2\) (this is consistent)
- Second row, first column: \(3 - y = y + 9\), solving for \(y\):
  \[
  3 - y = y + 9 \quad \Rightarrow \quad 3 - 9 = 2y \quad \Rightarrow \quad y = -3
  \]
- Second row, second column: \(z = z\) (this is consistent)

Thus, the solution for part (c) is:
\[
x = 6, \quad y = -3
\]

---

### Final Answers:
- **a**: \(x = 4\), \(y = 3\), \(z = -2\)
- **b**: \(x = 3\), \(y = 0\), \(z = 0 \text{ or } 1\)
- **c**: \(x = 6\), \(y = -3\)

Would you like further details on any of these steps or have additional questions?

Here are five related questions to deepen your understanding:
1. What is the general method for solving a system of matrix equations?
2. How would the solution change if the matrices were not square?
3. Can there be multiple solutions to a matrix equation, and why?
4. What does it mean for two matrices to be equal, in terms of their elements?
5. How can matrices be applied to solve real-world problems, such as systems of linear equations?

**Tip**: When solving matrix equations, always ensure each corresponding element is compared directly, and watch for consistent variable values.

Tentukan nilai x, y, dan z dari kesamaan matriks berikut:

Step-by-step solution for solving matrix equations to find values of x, y, and z. Learn how matrix equality works by matching corresponding elements in algebraic equations.

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