Question

Find all the minors and the cofactors of




.

Answer 1

To find the minors and cofactors of a matrix, let's assume we have a matrix \( A \). However, since you didn't provide an explicit matrix, I will demonstrate the process using a general \( 3 \times 3 \) matrix as an example:

Let
\[
A = \begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{pmatrix}
\]

### Minors

The **minor** \( M_{ij} \) of an element \( a_{ij} \) is defined as the determinant of the submatrix formed by deleting the \( i^{th} \) row and \( j^{th} \) column from \( A \).

#### Example minors:

- Minor \( M_{11} \) for element \( a_{11} \):
\[
M_{11} = \det\begin{pmatrix}
a_{22} & a_{23} \\
a_{32} & a_{33}
\end{pmatrix} = a_{22}a_{33} - a_{23}a_{32}
\]

- Minor \( M_{12} \) for element \( a_{12} \):
\[
M_{12} = \det\begin{pmatrix}
a_{21} & a_{23} \\
a_{31} & a_{33}
\end{pmatrix} = a_{21}a_{33} - a_{23}a_{31}
\]

- Minor \( M_{13} \) for element \( a_{13} \):
\[
M_{13} = \det\begin{pmatrix}
a_{21} & a_{22} \\
a_{31} & a_{32}
\end{pmatrix} = a_{21}a_{32} - a_{22}a_{31}
\]

Similarly, you can calculate the minors for \( M_{21} \), \( M_{22} \), \( M_{23} \), \( M_{31} \), \( M_{32} \), and \( M_{33} \).

### Cofactors

The **cofactor** \( C_{ij} \) of an element \( a_{ij} \) is defined as
\[
C_{ij} = (-1)^{i+j} M_{ij}
\]

#### Example cofactors:

- Cofactor \( C_{11} \):
\[
C_{11} = (-1)^{1+1} M_{11} = M_{11} = a_{22}a_{33} - a_{23}a_{32}
\]

- Cofactor \( C_{12} \):
\[
C_{12} = (-1)^{1+2} M_{12} = -M_{12} = -(a_{21}a_{33} - a_{23}a_{31})
\]

- Cofactor \( C_{13} \):
\[
C_{13} = (-1)^{1+3} M_{13} = M_{13} = a_{21}a_{32} - a_{22}a_{31}
\]

Again, you can proceed to calculate the cofactors for \( C_{21} \), \( C_{22} \), \( C_{23} \), \( C_{31} \), \( C_{32} \), and \( C_{33} \).

### Conclusion

The complete calculation of minors and cofactors will depend on the specific values in your matrix. If you provide an explicit matrix \( A \), I can compute the minors and cofactors for that specific case!