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

Options

A. `[{:(0,10),(10,0):}]`
B. `[{:(10,0),(0,10):}]`
C. `[{:(1,10),(10,1):}]`
D. `[{:(10,1),(1,10):}]`

Explanation

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

### Step 1: Calculate the Determinant of Matrix A
The determinant of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by the formula:

\[ \text{det}(A) = ad - bc \]

For our matrix \( A \):

\[ \text{det}(A) = (3)(4) - (2)(1) = 12 - 2 = 10 \]

### Step 2: Use the Property of A and adj A
The property states that:

\[ A \cdot \text{adj} A = \text{det}(A) \cdot I_n \]

where \( I_n \) is the identity matrix of order \( n \). Since our matrix \( A \) is 2x2, \( n = 2 \).

### Step 3: Write the Identity Matrix of Order 2
The identity matrix of order 2 is:

\[ I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

### Step 4: Multiply the Determinant by the Identity Matrix
Now, we multiply the determinant we found by the identity matrix:

\[ A \cdot \text{adj} A = \text{det}(A) \cdot I_2 = 10 \cdot \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 10 & 0 \\ 0 & 10 \end{pmatrix} \]

### Final Result
Thus, we conclude that:

\[ A \cdot \text{adj} A = \begin{pmatrix} 10 & 0 \\ 0 & 10 \end{pmatrix} \]