https://file.aiquickdraw.com/mathbot/file/1725476036461lobhlvkk.jpg

The text extraction from the image did not yield accurate results. However, I can clearly see the matrix equation in the image. Let's solve it step by step.

### Problem:
Given the equation:

\[
\begin{pmatrix}
-2 & 4 \\
6 & -8
\end{pmatrix}
\begin{pmatrix}
w & x \\
y & z
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
\]

We need to find the value of \( z \).

### Steps to solve:

1. Perform the matrix multiplication on the left-hand side.
2. Set the resulting matrix equal to the identity matrix.
3. Solve for \( z \).

Let's proceed with the calculations.The value of \( z \) is approximately \( \frac{1}{4} \). However, none of the given options include \( \frac{1}{4} \), so the correct answer is:

- **e. None of the options**

Would you like further details on how this solution was derived or have any other questions?

Here are five related questions you might find helpful:

1. How do you determine the inverse of a 2x2 matrix?
2. What properties of the identity matrix were used in this solution?
3. How do you verify the result of matrix multiplication?
4. What happens if a matrix is not invertible? How would you solve such a problem?
5. Can you solve for other variables like \( w \), \( x \), and \( y \)?

**Tip:** When solving matrix equations, always verify if the matrix is invertible before attempting to find its inverse.

If the matrix equation is given as: 
\[ \begin{pmatrix} -2 & 4 \\ 6 & -8 \end{pmatrix} \begin{pmatrix} w & x \\ y & z \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \] then what is the value of z?

Learn how to solve for z in a matrix equation using matrix multiplication and the properties of the identity matrix. This problem involves understanding linear algebra concepts such as matrix equations, identity matrices, and matrix multiplication, suitable for students in Grades 10-12.

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