Write the minors and co-factors of each elements of the first column of the matrix A `A=[(1,-3,2),(4,-1,2),(3,5,2)]`

Answer

To find the minors and cofactors of each element of the first column of the matrix \( A = \begin{pmatrix} 1 & -3 & 2 \\ 4 & -1 & 2 \\ 3 & 5 & 2 \end{pmatrix} \), we will follow these steps:

### Step 1: Identify the elements of the first column
The first column of matrix \( A \) consists of the elements:
- \( a_{11} = 1 \)
- \( a_{21} = 4 \)
- \( a_{31} = 3 \)

### Step 2: Calculate the minor of \( a_{11} \)
The minor \( M_{11} \) is the determinant of the submatrix obtained by removing the first row and first column:
\[ M_{11} = \begin{vmatrix} -1 & 2 \\ 5 & 2 \end{vmatrix} = (-1) \cdot 2 - (2) \cdot 5 = -2 - 10 = -12 \]

### Step 3: Calculate the cofactor of \( a_{11} \)
The cofactor \( C_{11} \) is given by:
\[ C_{11} = (-1)^{1+1} M_{11} = 1 \cdot (-12) = -12 \]

### Step 4: Calculate the minor of \( a_{21} \)
The minor \( M_{21} \) is the determinant of the submatrix obtained by removing the second row and first column:
\[ M_{21} = \begin{vmatrix} -3 & 2 \\ 5 & 2 \end{vmatrix} = (-3) \cdot 2 - (2) \cdot 5 = -6 - 10 = -16 \]

### Step 5: Calculate the cofactor of \( a_{21} \)
The cofactor \( C_{21} \) is given by:
\[ C_{21} = (-1)^{2+1} M_{21} = -1 \cdot (-16) = 16 \]

### Step 6: Calculate the minor of \( a_{31} \)
The minor \( M_{31} \) is the determinant of the submatrix obtained by removing the third row and first column:
\[ M_{31} = \begin{vmatrix} -3 & 2 \\ -1 & 2 \end{vmatrix} = (-3) \cdot 2 - (2) \cdot (-1) = -6 + 2 = -4 \]

### Step 7: Calculate the cofactor of \( a_{31} \)
The cofactor \( C_{31} \) is given by:
\[ C_{31} = (-1)^{3+1} M_{31} = 1 \cdot (-4) = -4 \]

### Summary of Results
- Minor of \( a_{11} \): \( M_{11} = -12 \), Cofactor \( C_{11} = -12 \)
- Minor of \( a_{21} \): \( M_{21} = -16 \), Cofactor \( C_{21} = 16 \)
- Minor of \( a_{31} \): \( M_{31} = -4 \), Cofactor \( C_{31} = -4 \)

### Final Answer
The minors and cofactors of the elements of the first column of matrix \( A \) are:
- For \( a_{11} \): \( M_{11} = -12, C_{11} = -12 \)
- For \( a_{21} \): \( M_{21} = -16, C_{21} = 16 \)
- For \( a_{31} \): \( M_{31} = -4, C_{31} = -4 \)