If A is 2-rowed square matrix and |A|=6 then A,adjA=?

Options

A. `[(6,0),(0,6)]`
B. `[(3,0),(0,3)]`
C. `[(1/6,0),(0,1/6)]`
D. none of these

Explanation

To solve the problem, we need to find the product of a matrix \( A \) and its adjugate \( \text{adj} A \), given that the determinant of \( A \) (denoted as \( |A| \)) is 6.

### Step-by-Step Solution:

1. **Understanding the Given Information:**
We know that \( A \) is a 2x2 matrix and \( |A| = 6 \).

2. **Using the Formula for the Product of a Matrix and its Adjugate:**
The relationship between a matrix \( A \) and its adjugate \( \text{adj} A \) is given by the formula:
\[ A \cdot \text{adj} A = |A| \cdot I \]
where \( I \) is the identity matrix of the same order as \( A \).

3. **Substituting the Determinant:**
Since we know \( |A| = 6 \), we can substitute this value into the formula:
\[ A \cdot \text{adj} A = 6 \cdot I \]

4. **Identifying the Identity Matrix:**
For a 2x2 matrix, the identity matrix \( I \) is:
\[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

5. **Calculating \( 6 \cdot I \):**
Now we multiply the identity matrix by 6:
\[ 6 \cdot I = 6 \cdot \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 6 & 0 \\ 0 & 6 \end{pmatrix} \]

6. **Final Result:**
Therefore, we conclude that:
\[ A \cdot \text{adj} A = \begin{pmatrix} 6 & 0 \\ 0 & 6 \end{pmatrix} \]

### Summary of the Solution:
The product of the matrix \( A \) and its adjugate \( \text{adj} A \) is:
\[ A \cdot \text{adj} A = \begin{pmatrix} 6 & 0 \\ 0 & 6 \end{pmatrix} \]