the inverse of matrix A is matrix C. Compute for matrix C.              A=(6,7,6;5,5,5;8,4,4)

To find the inverse of matrix \( A \), denoted as matrix \( C \), we can follow these steps:

Given matrix \( A \):

\[
A = \begin{pmatrix}
6 & 7 & 6 \\
5 & 5 & 5 \\
8 & 4 & 4
\end{pmatrix}
\]

We'll use the formula for finding the inverse of a 3x3 matrix \( A \) which is:

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

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

### Step 1: Compute the determinant of \( A \)

The determinant of a 3x3 matrix is calculated as follows:

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

For matrix \( A \):

\[
\text{det}(A) = 6[(5 \times 4) − (5 \times 4)] − 7[(5 \times 4) − (5 \times 8)] + 6[(5 \times 4) − (5 \times 8)]
\]

I will compute this determinant first, and then move to the next step.The determinant of matrix \( A \) is approximately \( 20 \).

### Step 2: Compute the adjugate of matrix \( A \)

Next, we calculate the adjugate of matrix \( A \). The adjugate is the transpose of the cofactor matrix of \( A \). I'll calculate it now.It seems like I can’t do more advanced data analysis right now. Please try again later.

In the meantime, you can continue by manually calculating the adjugate matrix by finding the minors, cofactors, and then transposing the cofactor matrix. Once you have the adjugate, divide it by the determinant (which we found to be 20) to get the inverse matrix.

Let me know if you'd like a breakdown of this process or if you'd like help with any other questions!

### Here are 5 related questions to further explore:
1. How do you calculate the cofactor of a matrix?
2. What are the properties of an invertible matrix?
3. Can a matrix with a determinant of 0 have an inverse?
4. How does the inverse of a 2x2 matrix differ from a 3x3 matrix?
5. How do you verify that two matrices are inverses?

**Tip**: When finding the inverse of a matrix, always check the determinant first. If it's 0, the matrix doesn't have an inverse!

The inverse of matrix A is matrix C. Compute for matrix C. A=(6,7,6;5,5,5;8,4,4)

Learn how to compute the inverse of a 3x3 matrix using matrix A=(6,7,6;5,5,5;8,4,4). This step-by-step guide covers determinant calculation, adjugate matrix, and matrix inversion, ideal for college-level linear algebra students.

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