If A is a non-singular matrix of order n, then `A(adj A)=`

Options

A. `A`
B. `I`
C. `|A|I_n`
D. `|A|^(2)I_n`

Answer

To solve the problem, we need to find the expression for \( A \cdot \text{adj} A \) where \( A \) is a non-singular matrix of order \( n \).

### Step-by-Step Solution:

1. **Understanding Non-Singular Matrix**:
A non-singular matrix \( A \) is one whose determinant is not equal to zero, i.e., \( \text{det}(A) \neq 0 \).

2. **Definition of Adjoint**:
The adjoint (or adjugate) of a matrix \( A \), denoted as \( \text{adj} A \), is defined such that:
\[ A \cdot \text{adj} A = \text{det}(A) \cdot I_n \]
where \( I_n \) is the identity matrix of order \( n \).

3. **Applying the Definition**:
Since \( A \) is non-singular, we can use the above property directly:
\[ A \cdot \text{adj} A = \text{det}(A) \cdot I_n \]

4. **Conclusion**:
Therefore, the expression \( A \cdot \text{adj} A \) simplifies to:
\[ A \cdot \text{adj} A = \text{det}(A) \cdot I_n \]

### Final Answer:
\[ A \cdot \text{adj} A = \text{det}(A) \cdot I_n \]