Let A be a square matrix of order `3xx3`, then `|k A|`is equal to(A) `k|A|` (B) `k^2|A|` (C) `k^3|A|` (D) `3k |A|`

Explanation

To solve the problem, we need to find the determinant of the matrix \( kA \), where \( A \) is a \( 3 \times 3 \) matrix and \( k \) is a scalar.

### Step-by-Step Solution:

1. **Define the Matrix \( A \)**:
Let \( A \) be a \( 3 \times 3 \) matrix represented as:
\[ A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \]

2. **Multiply the Matrix by Scalar \( k \)**:
The matrix \( kA \) is obtained by multiplying each element of \( A \) by the scalar \( k \):
\[ kA = \begin{pmatrix} k \cdot a_{11} & k \cdot a_{12} & k \cdot a_{13} \\ k \cdot a_{21} & k \cdot a_{22} & k \cdot a_{23} \\ k \cdot a_{31} & k \cdot a_{32} & k \cdot a_{33} \end{pmatrix} \]

3. **Apply the Determinant Property**:
The property of determinants states that if a matrix is multiplied by a scalar \( k \), the determinant of the resulting matrix is equal to \( k^n \) times the determinant of the original matrix, where \( n \) is the order of the matrix. For a \( 3 \times 3 \) matrix, \( n = 3 \):
\[ |kA| = k^3 |A| \]

4. **Conclusion**:
Therefore, the determinant of the matrix \( kA \) is given by:
\[ |kA| = k^3 |A| \]

5. **Identify the Correct Option**:
From the options provided, the correct answer is:
\[ (C) \, k^3 |A| \]