Let's solve the given matrix equation step-by-step.

The problem is as follows: $\begin{bmatrix} x + y & y + 2z \\ z - 1 & x \end{bmatrix} = \begin{bmatrix} 3 & 8 \\ 2 & 1 \end{bmatrix}$

We can equate corresponding elements from both matrices:

1. $x + y = 3$
2. $y + 2z = 8$
3. $z - 1 = 2$
4. $x = 1$

Let's solve these equations one by one:

Step 1: Solve $x = 1$

Directly from equation 4, $x = 1$.

Step 2: Substitute $x = 1$ into $x + y = 3$

$1 + y = 3$ $y = 2$

Step 3: Substitute $z - 1 = 2$

$z = 3$

Step 4: Verification using $y + 2z = 8$

Substituting $y = 2$ and $z = 3$: $2 + 2(3) = 2 + 6 = 8$ This is correct.

Hence, the values are: $x = 1, \quad y = 2, \quad z = 3$

Would you like any further details or have any questions about this solution?

Related Questions

1. What would happen if the matrix had different values? Could there still be a unique solution?
2. How can you verify if a matrix equation always has a solution?
3. What are the possible ways to solve systems of linear equations with matrices?
4. How does this problem relate to finding the inverse of a matrix?
5. What are the applications of solving matrix equations in real-world problems?

Tip

When solving matrix equations, always start by equating corresponding elements, which makes it easier to solve step-by-step.