https://file.aiquickdraw.com/mathbot/file/17270091238767xxxfu6c.jpg

The image shows a 3x3 matrix:

\[
\begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 8 \\
\end{bmatrix}
\]

It seems you're asked to find the inverse of this matrix. Let's go through the steps.

### Step 1: Calculate the Determinant of the Matrix
For a 3x3 matrix \( A \), the determinant is calculated as:

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

Where the matrix elements are:

\[
A = \begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 8 \\
\end{bmatrix}
\]

### Step 2: Check if the Determinant is Non-zero
If the determinant is non-zero, the matrix is invertible. Otherwise, it is not invertible.

Let me calculate the determinant for you.The determinant of the given matrix is approximately \( 3 \). Since the determinant is non-zero, the matrix is invertible.

### Step 3: Calculate the Inverse
Now that we know the matrix is invertible, the next step is to compute its inverse. Let me calculate it for you.The inverse of the given matrix is:

\[
\begin{bmatrix}
-2.67 & 2.67 & -1.00 \\
3.33 & -4.33 & 2.00 \\
-1.00 & 2.00 & -1.00 \\
\end{bmatrix}
\]

Would you like further explanation on how the inverse is calculated, or do you have any other questions?

Here are five related questions you might find useful:
1. What is the significance of the determinant in matrix inversion?
2. How can the adjugate matrix help in finding the inverse?
3. What happens if a matrix has a determinant of zero?
4. How is matrix inversion used in solving systems of linear equations?
5. What are the properties of an identity matrix?

**Tip:** The determinant also gives insight into whether a matrix transformation preserves the volume of a region in space (non-zero determinant) or collapses it (zero determinant).

Find the inverse of the following 3x3 matrix: 
[ [1, 2, 3], [4, 5, 6], [7, 8, 8] ]

Learn how to find the inverse of a 3x3 matrix using determinants and the adjugate matrix method. This example covers key concepts in linear algebra, including matrix inversion and determinant calculation. Suitable for high school and college-level students.

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