Let matrix B be the adjoint of a square matrix A, l be the identity matrix of the same order as A. If k (`ne`0) is the determinant of the matrix A, then what is AB equal to ?

Options

A. l
B. k l
C. `k^(2) I`
D. `(1//k)I`

Explanation

To solve the problem, we need to find the product of matrix A and its adjoint, which is denoted as B. Let's go through the steps systematically.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- Let \( A \) be a square matrix.
- Let \( B \) be the adjoint of matrix \( A \), i.e., \( B = \text{adj}(A) \).
- Let \( I \) be the identity matrix of the same order as \( A \).
- The determinant of matrix \( A \) is given as \( k \) (where \( k \neq 0 \)), i.e., \( \det(A) = k \).

2. **Using the Property of Adjoint**:
- A key property of matrices is that for any square matrix \( A \):
\[ A \cdot \text{adj}(A) = \det(A) \cdot I \]
- Applying this property to our matrices, we have:
\[ A \cdot B = A \cdot \text{adj}(A) = \det(A) \cdot I \]

3. **Substituting the Determinant**:
- Since we know that \( \det(A) = k \), we can substitute this into the equation:
\[ A \cdot B = k \cdot I \]

4. **Conclusion**:
- Therefore, we conclude that:
\[ AB = kI \]

### Final Answer:
Thus, the product \( AB \) is equal to \( kI \).

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