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.