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

Options

A. 20
B. 100
C. 10
D. 0

Answer

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. **Recall the property of matrices**: We know that for any invertible matrix \( A \), the product of \( A \) and its adjugate (adjoint) can be expressed as:
\[ A \cdot (\text{adj} \, A) = |A| \cdot I \]
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 equated to the expression we derived:
\[ |A| \cdot I = \begin{pmatrix} |A| & 0 \\ 0 & |A| \end{pmatrix} \]

3. **Compare the matrices**: Now, we can compare the two matrices:
\[ \begin{pmatrix} |A| & 0 \\ 0 & |A| \end{pmatrix} = \begin{pmatrix} 10 & 0 \\ 0 & 10 \end{pmatrix} \]

4. **Extract the determinant**: From the comparison, we see that:
\[ |A| = 10 \]

5. **Conclusion**: Therefore, the determinant of matrix \( A \) is:
\[ |A| = 10 \]

### Final Answer:
\[ |A| = 10 \]