If matrix `A=[(2,2),(2,3)]` then the value of [adj. A] equals to :

Options

A. 2
B. 3
C. 4
D. 6

Answer

To find the value of the adjoint of the matrix \( A = \begin{pmatrix} 2 & 2 \\ 2 & 3 \end{pmatrix} \), we will follow these steps:

### Step 1: Calculate the Determinant of Matrix A
The determinant of a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by the formula:
\[ \text{det}(A) = ad - bc \]
For our matrix \( A \):
- \( a = 2 \)
- \( b = 2 \)
- \( c = 2 \)
- \( d = 3 \)

Thus, the determinant is:
\[ \text{det}(A) = (2)(3) - (2)(2) = 6 - 4 = 2 \]

### Step 2: Use the Formula for the Determinant of the Adjoint of A
For a 2x2 matrix, the determinant of the adjoint of \( A \) can be calculated using the formula:
\[ \text{det}(\text{adj}(A)) = \text{det}(A)^{n-1} \]
where \( n \) is the order of the matrix. Here, \( n = 2 \).

Substituting the values we have:
\[ \text{det}(\text{adj}(A)) = \text{det}(A)^{2-1} = \text{det}(A)^{1} = \text{det}(A) \]
Since we calculated \( \text{det}(A) = 2 \), we have:
\[ \text{det}(\text{adj}(A)) = 2 \]

### Conclusion
Thus, the value of \( \text{adj}(A) \) is such that its determinant is equal to 2. Therefore, the answer is:
\[ \text{det}(\text{adj}(A)) = 2 \]