Let A be a square matrix of order 3 such that `|Adj A | =100 ` then |A| equals

Options

A. `pm 10`
B. `-100`
C. 100
D. 25

Answer

To solve the problem, we need to find the determinant of the square matrix \( A \) given that the determinant of the adjoint of \( A \) is \( | \text{Adj} A | = 100 \).

### Step-by-step Solution:

1. **Understanding the relationship between the determinant of a matrix and its adjoint**:
The determinant of the adjoint of a matrix \( A \) of order \( n \) is given by:
\[ | \text{Adj} A | = |A|^{n-1} \]
For a square matrix \( A \) of order 3, this becomes:
\[ | \text{Adj} A | = |A|^{3-1} = |A|^2 \]

2. **Setting up the equation**:
Given that \( | \text{Adj} A | = 100 \), we can write:
\[ |A|^2 = 100 \]

3. **Solving for \( |A| \)**:
To find \( |A| \), we take the square root of both sides:
\[ |A| = \sqrt{100} \]
This gives us:
\[ |A| = 10 \quad \text{or} \quad |A| = -10 \]

4. **Conclusion**:
Therefore, the determinant \( |A| \) can be either \( 10 \) or \( -10 \).

### Final Answer:
\[ |A| = 10 \quad \text{or} \quad |A| = -10 \]