Find the adjoint of matrix`[{:(1,2),(3,7):}]`

Options

A. `[{:(-7,-2),(-3," "1):}]`
B. `[{:(7,-2),(3," "1):}]`
C. `[{:(" "7,-2),(-3,-1):}]`
D. `[{:(" "7,-2),(-3," "1):}]`

Answer

To find the adjoint of the given matrix \( A = \begin{pmatrix} 1 & 2 \\ 3 & 7 \end{pmatrix} \), we will follow the steps outlined below:

### Step 1: Identify the elements of the matrix
The given 2x2 matrix is:
\[ A = \begin{pmatrix} 1 & 2 \\ 3 & 7 \end{pmatrix} \]
Here, the elements are:
- \( a = 1 \)
- \( b = 2 \)
- \( c = 3 \)
- \( d = 7 \)

### Step 2: Apply 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
Substituting the values of \( a, b, c, \) and \( d \) into the formula:
- \( d = 7 \)
- \( -b = -2 \)
- \( -c = -3 \)
- \( a = 1 \)

Thus, we have:
\[ \text{adj}(A) = \begin{pmatrix} 7 & -2 \\ -3 & 1 \end{pmatrix} \]

### Step 4: Write the final result
The adjoint of the matrix \( A \) is:
\[ \text{adj}(A) = \begin{pmatrix} 7 & -2 \\ -3 & 1 \end{pmatrix} \]

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