If `A[(-1,5),(-3,2)]`, then adj A=

Options

A. `[(2,3),(-5,-1)]`
B. `[(1,-5),(3,-2)]`
C. `[(2,-5),(3,-1)]`
D. `[(-2,5),(-3,1)]`

Answer

To find the adjoint of the matrix \( A = \begin{pmatrix} -1 & 5 \\ -3 & 2 \end{pmatrix} \), we will follow the steps outlined below:

### Step 1: Identify the elements of the matrix
The matrix \( A \) is a 2x2 matrix with elements:
- \( a = -1 \)
- \( b = 5 \)
- \( c = -3 \)
- \( d = 2 \)

### Step 2: Use the formula for the adjoint of a 2x2 matrix
The adjoint of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:
\[ \text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]

### Step 3: Substitute the values into the formula
Now, substituting the values of \( a \), \( b \), \( c \), and \( d \):
- \( d = 2 \)
- \( -b = -5 \) (since \( b = 5 \))
- \( -c = 3 \) (since \( c = -3 \))
- \( a = -1 \)

So we can write:
\[ \text{adj}(A) = \begin{pmatrix} 2 & -5 \\ 3 & -1 \end{pmatrix} \]

### Step 4: Write the final answer
Thus, the adjoint of the matrix \( A \) is:
\[ \text{adj}(A) = \begin{pmatrix} 2 & -5 \\ 3 & -1 \end{pmatrix} \]

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