Find the inverse of the matrix (if it exists). (If an answer does not exist, enter DNE.)

−3	−2	6	

4	2	−6
6	3	−8


We are tasked with finding the inverse of the matrix:

\[
A = \begin{pmatrix}
-3 & -2 & 6 \\
4 & 2 & -6 \\
6 & 3 & -8
\end{pmatrix}
\]

To find the inverse of a matrix, we need to check if the determinant of the matrix is non-zero. If the determinant is zero, the inverse does not exist (DNE).

The determinant of a 3x3 matrix:

\[
A = \begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix}
\]

is calculated as:

\[
\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
\]

### Step 1: Calculate the determinant of matrix A
Using the given matrix values:
\[
\text{det}(A) = (-3) \left( (2)(-8) - (-6)(3) \right) - (-2) \left( (4)(-8) - (-6)(6) \right) + (6) \left( (4)(3) - (2)(6) \right)
\]

Let's compute this step by step.

### Step 2: Check if determinant is zero

Let me compute the determinant.The determinant of the matrix is \( 2 \), which is non-zero. Therefore, the inverse of the matrix exists.

### Step 3: Find the inverse of the matrix

The inverse of a 3x3 matrix \( A \) is given by:

\[
A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)
\]

Where \( \text{adj}(A) \) is the adjugate (or adjoint) of the matrix, which is the transpose of the cofactor matrix. Let me now calculate the inverse of this matrix.

It seems like I can't do more advanced data analysis right now. Please try again later. However, I can guide you through the process of calculating the inverse manually if you'd like!

Would you like to continue with the manual steps?

In the meantime, here are 5 related questions to explore:
1. What conditions determine whether a matrix is invertible?
2. How is the adjugate of a matrix related to the cofactor matrix?
3. What are some practical applications of finding matrix inverses?
4. Can the determinant of a matrix be negative? What does that imply?
5. How would you find the inverse of a 2x2 matrix?

**Tip**: The inverse of a matrix can be used to solve systems of linear equations via matrix multiplication.

Find the inverse of the matrix (if it exists). (If an answer does not exist, enter DNE.)

−3 −2 6
4 2 −6
6 3 −8

Explore how to find the inverse of a 3x3 matrix, including step-by-step determinant calculation and the use of adjugate matrices. The matrix provided is -3 -2 6, 4 2 -6, 6 3 -8. This guide is suitable for advanced high school or college-level students studying linear algebra.

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