For a invertible matrix A if `A(adjA)=[(10,0),(0,10)]`, then |A|=

Options

A. 100
B. `-100`
C. 10
D. `-10`

Explanation

To solve the problem, we need to find the determinant of the invertible matrix \( A \) given that \( A \cdot \text{adj}(A) = \begin{pmatrix} 10 & 0 \\ 0 & 10 \end{pmatrix} \).

### Step-by-Step Solution:

1. **Understand the relationship**: We know that for any square matrix \( A \), the product of \( A \) and its adjugate (or adjoint) \( \text{adj}(A) \) is given by the formula:
\[ A \cdot \text{adj}(A) = |A| I \]
where \( |A| \) is the determinant of \( A \) and \( I \) is the identity matrix.

2. **Set up the equation**: From the problem, we have:
\[ A \cdot \text{adj}(A) = \begin{pmatrix} 10 & 0 \\ 0 & 10 \end{pmatrix} \]
This can be rewritten using the formula:
\[ |A| I = \begin{pmatrix} 10 & 0 \\ 0 & 10 \end{pmatrix} \]

3. **Identify the identity matrix**: The identity matrix \( I \) for a 2x2 matrix is:
\[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

4. **Express the equation**: Therefore, we can express the equation as:
\[ |A| \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 10 & 0 \\ 0 & 10 \end{pmatrix} \]

5. **Compare the matrices**: This implies:
\[ |A| \cdot 1 = 10 \]
Therefore, we have:
\[ |A| = 10 \]

### Conclusion:
The determinant of the matrix \( A \) is:
\[ |A| = 10 \]