https://file.aiquickdraw.com/mathbot/file/1726618682543mrxek86d.jpg

The matrix provided in the image is a 5x5 matrix. To find the inverse of this matrix, we need to follow a process that involves several steps. Let’s break it down:

### Given Matrix:
\[
\begin{bmatrix}
80 & 71 & 62 & 91 & 51 \\
81 & 96 & 82 & 72 & 95 \\
75 & 81 & 87 & 83 & 70 \\
85 & 88 & 76 & 89 & 74 \\
74 & 64 & 91 & 50 & 73
\end{bmatrix}
\]

### Steps to calculate the inverse:

1. **Check if the matrix is invertible**: 
   We first calculate the determinant of the matrix. If the determinant is zero, the matrix is not invertible.

2. **Adjugate matrix**: 
   If the determinant is non-zero, the inverse of the matrix is calculated as the product of the reciprocal of the determinant and the adjugate matrix (the transpose of the cofactor matrix).

3. **Inverse formula**: 
   The inverse of matrix \(A\) is given by:
   \[
   A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)
   \]
   where \(\text{det}(A)\) is the determinant and \(\text{adj}(A)\) is the adjugate matrix.

Let me calculate the determinant first, then we can proceed to compute the inverse matrix if the determinant is non-zero. I'll do this now.The determinant of the given matrix is approximately \(-1,747,039\), which is non-zero. This means the matrix is invertible, and we can proceed to calculate the inverse.

Let me compute the inverse of the matrix now.The inverse of the given matrix is:

\[
\begin{bmatrix}
-0.1176 & -0.1223 & -0.1016 & 0.2872 & 0.0476 \\
-0.6732 & -0.5011 & -0.1258 & 1.1908 & 0.0359 \\
-0.1114 & -0.0950 & 0.0199 & 0.1620 & 0.0182 \\
0.3897 & 0.2940 & 0.1108 & -0.6987 & -0.0528 \\
0.5813 & 0.4804 & 0.1126 & -1.0585 & -0.0526
\end{bmatrix}
\]

If you have any questions or would like further details on how the inverse was calculated, feel free to ask!

### Related Questions:
1. How do you calculate the determinant of a matrix manually?
2. What are the conditions under which a matrix is non-invertible?
3. How is the adjugate matrix related to finding the inverse of a matrix?
4. Can we find the inverse of a matrix using row reduction?
5. What are some real-world applications of matrix inversion?

### Tip:
Inverting large matrices manually can be complex and prone to errors. Using computational tools like Python or a calculator is highly recommended for such tasks.

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Learn how to find the inverse of a 5x5 matrix with a detailed step-by-step solution. This problem includes calculating the determinant, adjugate matrix, and applying the matrix inversion formula. Suitable for advanced high school or college-level students.

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