If `A=[[3, 4], [5, 7]]`, then A(adjA)=

Options

A. `|A|I`
B. `|A|`
C. `I`
D. `2I`

Answer

To solve the problem \( A \cdot \text{adj}(A) \) where \( A = \begin{bmatrix} 3 & 4 \\ 5 & 7 \end{bmatrix} \), we can use the formula:

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

where \( I_n \) is the identity matrix of the same order as \( A \).

### Step 1: Calculate the Determinant of A

The determinant of a \( 2 \times 2 \) matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is given by:

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

For our matrix \( A = \begin{bmatrix} 3 & 4 \\ 5 & 7 \end{bmatrix} \):

- \( a = 3 \)
- \( b = 4 \)
- \( c = 5 \)
- \( d = 7 \)

Now, substituting these values into the determinant formula:

\[ \text{det}(A) = (3 \cdot 7) - (4 \cdot 5) = 21 - 20 = 1 \]

### Step 2: Write the Identity Matrix \( I_2 \)

The identity matrix \( I_2 \) for a \( 2 \times 2 \) matrix is:

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

### Step 3: Calculate \( A \cdot \text{adj}(A) \)

Using the formula \( A \cdot \text{adj}(A) = \text{det}(A) \cdot I_n \):

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

### Final Result

Thus, we have:

\[ A \cdot \text{adj}(A) = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \]