To find the minors of a matrix, we calculate the determinant of each \(2 \times 2\) submatrix formed by removing one row and one column from the original \(3 \times 3\) matrix. Each minor corresponds to a specific element in the matrix.

Given the matrix \( A \):

\[
A = \begin{pmatrix}
1 & 2 & -1 \\
2 & 1 & 2 \\
-1 & 2 & 1
\end{pmatrix}
\]

The minor \( M_{ij} \) corresponds to the element at the \( i \)-th row and \( j \)-th column of the matrix. To calculate each minor:

### Minor \( M_{11} \):
Remove the first row and first column:

\[
M_{11} = \text{det} \begin{pmatrix}
1 & 2 \\
2 & 1
\end{pmatrix}
\]

\[
M_{11} = (1)(1) - (2)(2) = 1 - 4 = -3
\]

### Minor \( M_{12} \):
Remove the first row and second column:

\[
M_{12} = \text{det} \begin{pmatrix}
2 & 2 \\
-1 & 1
\end{pmatrix}
\]

\[
M_{12} = (2)(1) - (2)(-1) = 2 + 2 = 4
\]

### Minor \( M_{13} \):
Remove the first row and third column:

\[
M_{13} = \text{det} \begin{pmatrix}
2 & 1 \\
-1 & 2
\end{pmatrix}
\]

\[
M_{13} = (2)(2) - (1)(-1) = 4 + 1 = 5
\]

### Minor \( M_{21} \):
Remove the second row and first column:

\[
M_{21} = \text{det} \begin{pmatrix}
2 & -1 \\
2 & 1
\end{pmatrix}
\]

\[
M_{21} = (2)(1) - (-1)(2) = 2 + 2 = 4
\]

### Minor \( M_{22} \):
Remove the second row and second column:

\[
M_{22} = \text{det} \begin{pmatrix}
1 & -1 \\
-1 & 1
\end{pmatrix}
\]

\[
M_{22} = (1)(1) - (-1)(-1) = 1 - 1 = 0
\]

### Minor \( M_{23} \):
Remove the second row and third column:

\[
M_{23} = \text{det} \begin{pmatrix}
1 & 2 \\
-1 & 2
\end{pmatrix}
\]

\[
M_{23} = (1)(2) - (2)(-1) = 2 + 2 = 4
\]

### Minor \( M_{31} \):
Remove the third row and first column:

\[
M_{31} = \text{det} \begin{pmatrix}
2 & -1 \\
1 & 2
\end{pmatrix}
\]

\[
M_{31} = (2)(2) - (-1)(1) = 4 + 1 = 5
\]

### Minor \( M_{32} \):
Remove the third row and second column:

\[
M_{32} = \text{det} \begin{pmatrix}
1 & -1 \\
2 & 2
\end{pmatrix}
\]

\[
M_{32} = (1)(2) - (-1)(2) = 2 + 2 = 4
\]

### Minor \( M_{33} \):
Remove the third row and third column:

\[
M_{33} = \text{det} \begin{pmatrix}
1 & 2 \\
2 & 1
\end{pmatrix}
\]

\[
M_{33} = (1)(1) - (2)(2) = 1 - 4 = -3
\]

So, the minors of matrix \( A \) are:

\[
\text{Minors of } A = \begin{pmatrix}
-3 & 4 & 5 \\
4 & 0 & 4 \\
5 & 4 & -3
\end{pmatrix}
\]

Would you like more details or have any other questions?

Here are 8 follow-up questions that you might find helpful:

1. How do minors relate to cofactors?
2. What is the significance of the cofactor matrix?
3. How can minors be used to compute the determinant of a larger matrix?
4. What is the difference between minors and cofactors?
5. How can minors be used to find the inverse of a matrix?
6. How are minors used in calculating the adjugate of a matrix?
7. Can minors be used to simplify matrix determinants for large matrices?
8. How do minors contribute to the calculation of eigenvalues?

**Tip:** When calculating minors, it's helpful to systematically remove one row and one column to ensure accurate results. Understanding minors is crucial for advanced topics like finding determinants, inverses, and solving linear systems.

Learn how to find the minors of a matrix with a detailed step-by-step calculation. Understand the concept of minors and their application in matrix operations. Suitable for advanced high school students and above.

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