If `A=[{:(3,2),(1,4):}]` then A. adj A is equal to :

Options

A. `[{:(10,0),(0,10):}]`
B. `[{:(0,10),(10,0):}]`
C. `[{:(10,10),(0,0):}]`
D. `[{:(0,0),(10,10):}]`

Answer

To solve the problem of finding \( A \cdot \text{adj}(A) \) for the matrix \( A = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \), we will follow these steps:

### Step 1: Find the Adjoint of Matrix A

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

For our matrix \( A = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \):
- \( a = 3 \)
- \( b = 2 \)
- \( c = 1 \)
- \( d = 4 \)

Thus, the adjoint of \( A \) is:
\[ \text{adj}(A) = \begin{pmatrix} 4 & -2 \\ -1 & 3 \end{pmatrix} \]

### Step 2: Multiply Matrix A by its Adjoint

Now we need to compute \( A \cdot \text{adj}(A) \):
\[ A \cdot \text{adj}(A) = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \cdot \begin{pmatrix} 4 & -2 \\ -1 & 3 \end{pmatrix} \]

We will perform matrix multiplication as follows:

- **Element (1,1)**:
\[ 3 \cdot 4 + 2 \cdot (-1) = 12 - 2 = 10 \]

- **Element (1,2)**:
\[ 3 \cdot (-2) + 2 \cdot 3 = -6 + 6 = 0 \]

- **Element (2,1)**:
\[ 1 \cdot 4 + 4 \cdot (-1) = 4 - 4 = 0 \]

- **Element (2,2)**:
\[ 1 \cdot (-2) + 4 \cdot 3 = -2 + 12 = 10 \]

Putting it all together, we have:
\[ A \cdot \text{adj}(A) = \begin{pmatrix} 10 & 0 \\ 0 & 10 \end{pmatrix} \]

### Final Answer
Thus, the result of \( A \cdot \text{adj}(A) \) is:
\[ \begin{pmatrix} 10 & 0 \\ 0 & 10 \end{pmatrix} \]

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