If `A=[{:(2,-1," 3"),(4," 2"," 5"),(0," 4",-1):}]` , then cofactor `A_(32)` is

Options

A. `-2`
B. `-8`
C. 4
D. 2

Answer

To find the cofactor \( A_{32} \) of the matrix \( A \), we will follow these steps:

Given matrix:
\[ A = \begin{pmatrix} 2 & -1 & 3 \\ 4 & 2 & 5 \\ 0 & 4 & -1 \end{pmatrix} \]

### Step 1: Identify the element for which we need the cofactor
We need to find the cofactor \( A_{32} \), which corresponds to the element in the 3rd row and 2nd column of the matrix \( A \).

### Step 2: Calculate the minor \( M_{32} \)
The minor \( M_{32} \) is the determinant of the submatrix obtained by deleting the 3rd row and 2nd column from \( A \). The remaining elements are:
\[ \begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix} \]

### Step 3: Calculate the determinant of the submatrix
The determinant of a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is calculated as \( ad - bc \).

For our submatrix:
\[ \text{det} = (2)(5) - (3)(4) = 10 - 12 = -2 \]

Thus, \( M_{32} = -2 \).

### Step 4: Calculate the cofactor \( A_{32} \)
The cofactor \( A_{32} \) is given by:
\[ A_{32} = (-1)^{3+2} M_{32} \]
Since \( 3 + 2 = 5 \) is odd, \( (-1)^{5} = -1 \).

Now substituting the value of the minor:
\[ A_{32} = -1 \times (-2) = 2 \]

### Final Answer
The cofactor \( A_{32} \) is \( 2 \).

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