The adjoint of matrix `[(1,1),(5,7)]` is:

Options

A. `[(7,5),(1,1)]`
B. `[(-1,5),(1,-7)]`
C. `[(7,-1),(-5,1)]`
D. `[(7,-5),(-1,1)]`

Answer

To find the adjoint of the matrix \(\begin{pmatrix} 1 & 1 \\ 5 & 7 \end{pmatrix}\), we can follow these steps:

### Step 1: Identify the elements of the matrix
The given matrix is:
\[ A = \begin{pmatrix} 1 & 1 \\ 5 & 7 \end{pmatrix} \]
Here, we can identify:
- \(a = 1\)
- \(b = 1\)
- \(c = 5\)
- \(d = 7\)

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

### Step 3: Substitute the values into the formula
Substituting the identified values into the adjoint formula:
- \(d = 7\)
- \(-b = -1\)
- \(-c = -5\)
- \(a = 1\)

Thus, we get:
\[ \text{adj}(A) = \begin{pmatrix} 7 & -1 \\ -5 & 1 \end{pmatrix} \]

### Step 4: Write the final answer
The adjoint of the matrix \(\begin{pmatrix} 1 & 1 \\ 5 & 7 \end{pmatrix}\) is:
\[ \text{adj}(A) = \begin{pmatrix} 7 & -1 \\ -5 & 1 \end{pmatrix} \]

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