The co-factor of -4 and 9 in `[[-1, -2, 3], [-4, -5, -6], [-7, 8, 9]]` are respectively.

Options

A. `-42, 3`
B. `42, -3`
C. `-42, -3`
D. `42, 3`

Answer

To find the co-factors of the elements -4 and 9 in the matrix

\[ A = \begin{bmatrix} -1 & -2 & 3 \\ -4 & -5 & -6 \\ -7 & 8 & 9 \end{bmatrix} \]

we will calculate the co-factors \(C_{21}\) (for -4) and \(C_{33}\) (for 9).

### Step 1: Identify the co-factor positions
- The element -4 is located at position (2, 1), so we need to find \(C_{21}\).
- The element 9 is located at position (3, 3), so we need to find \(C_{33}\).

### Step 2: Calculate \(C_{21}\)
The co-factor \(C_{21}\) is given by:

\[ C_{21} = (-1)^{2+1} \cdot \text{det}(M_{21}) \]

where \(M_{21}\) is the minor obtained by deleting the 2nd row and 1st column from matrix \(A\):

\[ M_{21} = \begin{bmatrix} -2 & 3 \\ 8 & 9 \end{bmatrix} \]

Now, we calculate the determinant of \(M_{21}\):

\[ \text{det}(M_{21}) = (-2)(9) - (3)(8) = -18 - 24 = -42 \]

Thus,

\[ C_{21} = (-1)^{3} \cdot (-42) = -(-42) = 42 \]

### Step 3: Calculate \(C_{33}\)
The co-factor \(C_{33}\) is given by:

\[ C_{33} = (-1)^{3+3} \cdot \text{det}(M_{33}) \]

where \(M_{33}\) is the minor obtained by deleting the 3rd row and 3rd column from matrix \(A\):

\[ M_{33} = \begin{bmatrix} -1 & -2 \\ -4 & -5 \end{bmatrix} \]

Now, we calculate the determinant of \(M_{33}\):

\[ \text{det}(M_{33}) = (-1)(-5) - (-2)(-4) = 5 - 8 = -3 \]

Thus,

\[ C_{33} = (-1)^{6} \cdot (-3) = 1 \cdot (-3) = -3 \]

### Final Result
The co-factors of -4 and 9 are:

\[ C_{21} = 42 \quad \text{and} \quad C_{33} = -3 \]

### Summary
The co-factors of -4 and 9 in the matrix are \(42\) and \(-3\) respectively.

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