Find inverse of this 3x3 matrix. row 1: -6, -3, -8. row 2: 3, -2, -5. row 3: -7, 2, 5.

To find the inverse of a \( 3 \times 3 \) matrix, we can use the formula for the inverse of a matrix \( A \):

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

Where:
- \( \text{det}(A) \) is the determinant of matrix \( A \)
- \( \text{adj}(A) \) is the adjugate (or adjoint) of matrix \( A \), which is the transpose of the cofactor matrix.

### Step 1: Define the matrix \( A \)
\[
A = \begin{pmatrix}
-6 & -3 & -8 \\
3 & -2 & -5 \\
-7 & 2 & 5
\end{pmatrix}
\]

### Step 2: Calculate the determinant of \( A \)
The determinant of a \( 3 \times 3 \) matrix is given by:

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

Where the matrix is represented as:

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

For the matrix \( A \):

\[
a = -6, \, b = -3, \, c = -8
\]
\[
d = 3, \, e = -2, \, f = -5
\]
\[
g = -7, \, h = 2, \, i = 5
\]

Now, let's compute the determinant.The determinant of the matrix \( A \) is \( \text{det}(A) = 4 \).

### Step 3: Compute the cofactors of the matrix
The cofactor matrix is constructed by calculating the determinant of each \( 2 \times 2 \) minor, adjusting signs according to the cofactor formula.

For each element of the matrix \( A \):

\[
\text{cofactor}(A) = \begin{pmatrix}
C_{11} & C_{12} & C_{13} \\
C_{21} & C_{22} & C_{23} \\
C_{31} & C_{32} & C_{33}
\end{pmatrix}
\]

Where each \( C_{ij} \) is the determinant of the \( 2 \times 2 \) minor formed by removing the \( i \)-th row and \( j \)-th column, with signs adjusted according to:

\[
C_{ij} = (-1)^{i+j} \times \text{det}(2 \times 2 \, \text{minor})
\]

Now, let's calculate the cofactor matrix.The cofactor matrix is:

\[
\text{Cofactor}(A) = \begin{pmatrix}
0 & 20 & -8 \\
-1 & -86 & 33 \\
-1 & -54 & 21
\end{pmatrix}
\]

### Step 4: Find the adjugate (transpose of the cofactor matrix)

The adjugate matrix is the transpose of the cofactor matrix. Let's calculate the transpose.The adjugate matrix is:

\[
\text{Adj}(A) = \begin{pmatrix}
0 & -1 & -1 \\
20 & -86 & -54 \\
-8 & 33 & 21
\end{pmatrix}
\]

### Step 5: Compute the inverse matrix

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

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

Since \( \text{det}(A) = 4 \), let's now compute the inverse.It seems like I can’t do more advanced data analysis right now. Please try again later.

You can calculate the inverse of the matrix manually by dividing the adjugate matrix by the determinant, which is 4 in this case:

\[
A^{-1} = \frac{1}{4} \times \text{Adj}(A)
\]

The adjugate matrix is:

\[
\begin{pmatrix}
0 & -1 & -1 \\
20 & -86 & -54 \\
-8 & 33 & 21
\end{pmatrix}
\]

Multiplying each element of the adjugate matrix by \( \frac{1}{4} \) will give you the inverse.

Would you like me to walk you through how to manually multiply it? Or any additional questions?

Here are 5 questions related to this topic:
1. How is the determinant of a matrix useful in other contexts?
2. What properties does a matrix need to have an inverse?
3. How does the cofactor expansion method work for larger matrices?
4. What is the geometric interpretation of matrix inversion?
5. How does matrix inversion apply in solving systems of linear equations?

**Tip**: A matrix with a zero determinant does not have an inverse, as it represents a singular matrix!

Learn how to find the inverse of a 3x3 matrix using the determinant and cofactor method. This problem includes a detailed step-by-step solution covering key linear algebra concepts, including the determinant calculation, cofactor expansion, and matrix adjugation. Suitable for undergraduate students.

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