https://file.aiquickdraw.com/mathbot/file/17260574583601g7elbuz.jpg

To solve for \( z \) using Cramer's Rule, we need to substitute the third column of matrix \( A \) with the values of matrix \( B \), while leaving the rest of the matrix unchanged.

Given the matrix equation \( AX = B \), and the matrices:
\[
A = \begin{pmatrix} 1 & -1 & 2 \\ 1 & 1 & -3 \\ 2 & 3 & 1 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad B = \begin{pmatrix} 3 \\ -2 \\ 5 \end{pmatrix}, \quad |A| = 12
\]

### Step 1: Cramer's Rule for \( z \)

Cramer's rule states that:
\[
z = \frac{|A_z|}{|A|}
\]
Where \( A_z \) is the matrix formed by replacing the third column of \( A \) with \( B \).

### Step 2: Construct \( A_z \)

To construct \( A_z \), replace the third column of \( A \) with the vector \( B \):
\[
A_z = \begin{pmatrix} 1 & -1 & 3 \\ 1 & 1 & -2 \\ 2 & 3 & 5 \end{pmatrix}
\]

### Step 3: Determine the Determinant of \( A_z \)

Now, we compute the determinant of \( A_z \):
\[
\text{det}(A_z) = \begin{vmatrix} 1 & -1 & 3 \\ 1 & 1 & -2 \\ 2 & 3 & 5 \end{vmatrix}
\]

The correct form matches **Option 4** in the image.

Would you like a detailed calculation for the determinant or further clarification on any steps?

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Here are 5 related questions for further exploration:
1. How does Cramer's Rule work for systems with no unique solutions?
2. What happens if \( |A| = 0 \) in Cramer's Rule?
3. How do we calculate the determinants for higher-dimensional matrices?
4. Can Cramer's Rule be used efficiently for larger systems of equations?
5. How is matrix inversion related to Cramer's Rule?

**Tip**: Cramer's Rule is computationally expensive for larger matrices. For systems larger than 3x3, other methods like Gaussian elimination or matrix factorization are usually more efficient.

From the matrix form AX = B, given A = [ [1, -1, 2], [1, 1, -3], [2, 3, 1] ], B = [3, -2, 5], and |A| = 12, find z using Cramer's Rule.

Learn how to find z in a system of linear equations using Cramer's Rule. This example involves a 3x3 matrix, with detailed determinant calculations. Suitable for students in Grades 11-12 and college level.

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