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The image displays a problem asking for the inverse of matrix \( A \) using the cofactor minor method. Matrix \( A \) is given as:

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

To find the inverse of matrix \( A \) using the cofactor method, we follow these steps:

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

The determinant of a 3x3 matrix is calculated as:

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

Where:

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

In our case:

\[
a = 4, \, b = 2, \, c = 8 \\
d = 2, \, e = 1, \, f = 5 \\
g = 3, \, h = 2, \, i = 4
\]

Substitute into the formula:

\[
\text{det}(A) = 4[(1)(4) - (5)(2)] - 2[(2)(4) - (5)(3)] + 8[(2)(2) - (1)(3)]
\]

### Step 2: Compute the cofactor matrix

Once the determinant is calculated, the next step is to find the cofactor matrix of \( A \), which involves calculating the minors for each element of the matrix, followed by adjusting the signs based on the cofactor pattern.

### Step 3: Transpose the cofactor matrix

This gives us the adjugate (adjoint) matrix, which is the transpose of the cofactor matrix.

### Step 4: Calculate the inverse

Finally, the inverse of \( A \) is given by:

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

Let me calculate the determinant and proceed with the steps.The determinant of matrix \( A \) is \(-2\).

Since the determinant is non-zero, the matrix is invertible. Now we can proceed with the next steps to find the inverse using the cofactor method.

### Step 2: Compute the cofactor matrix
I will calculate the cofactor matrix next, then find the adjugate matrix and, finally, the inverse matrix.

Let me calculate that for you.It seems like I can't do more advanced data analysis right now. Please try again later. However, you can proceed with the following steps to complete the process manually:

1. **Calculate the cofactor matrix**: For each element of the matrix, calculate its minor (determinant of the 2x2 matrix formed by removing the corresponding row and column) and apply the appropriate sign (based on the cofactor pattern).

2. **Transpose the cofactor matrix**: This gives the adjugate matrix.

3. **Find the inverse**: Multiply the adjugate matrix by \( \frac{1}{\text{det}(A)} \), which is \( \frac{1}{-2} \) in this case.

If you'd like, I can guide you through the manual process or try again later.

Let me know if you have any questions, and here are some related questions to explore:

1. What is the significance of the determinant in finding an inverse matrix?
2. How do you compute minors and cofactors for larger matrices?
3. What are some applications of the cofactor method in real-world problems?
4. Why is matrix inversion important in solving linear systems?
5. How do we interpret the result if the determinant is zero?

**Tip:** Always check the determinant of a matrix before attempting to find its inverse, as a zero determinant means the matrix is non-invertible.

Find the inverse of matrix A using the cofactor minor method where A = [[4, 2, 8], [2, 1, 5], [3, 2, 4]].

Learn how to find the inverse of a 3x3 matrix A = [[4, 2, 8], [2, 1, 5], [3, 2, 4]] using the cofactor minor method. This problem involves calculating the determinant, cofactor matrix, adjugate matrix, and applying the inverse formula. Suitable for Grades 10-12.

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