If `A` is a `3xx3` matrix and `det (3A) = k det(A) , k` is equal to:

Options

A. (a) `9`
B. (b) `6`
C. (c) `1`
D. (d) `27`

Answer

To solve the problem, we need to find the value of \( k \) such that \( \text{det}(3A) = k \cdot \text{det}(A) \) for a \( 3 \times 3 \) matrix \( A \).

### Step-by-step Solution:

1. **Understanding Scalar Multiplication of Matrices**:
When we multiply a matrix \( A \) by a scalar (in this case, 3), each element of the matrix is multiplied by that scalar. Thus, if \( A \) is a \( 3 \times 3 \) matrix, then:
\[ 3A = \begin{pmatrix} 3a_{11} & 3a_{12} & 3a_{13} \\ 3a_{21} & 3a_{22} & 3a_{23} \\ 3a_{31} & 3a_{32} & 3a_{33} \end{pmatrix} \]

2. **Using the Property of Determinants**:
A key property of determinants states that if you multiply a matrix by a scalar \( c \), the determinant of the resulting matrix is given by:
\[ \text{det}(cA) = c^n \cdot \text{det}(A) \]
where \( n \) is the order of the matrix. For a \( 3 \times 3 \) matrix, \( n = 3 \).

3. **Applying the Property**:
In our case, we have:
\[ \text{det}(3A) = 3^3 \cdot \text{det}(A) \]
Calculating \( 3^3 \):
\[ 3^3 = 27 \]
Therefore:
\[ \text{det}(3A) = 27 \cdot \text{det}(A) \]

4. **Identifying \( k \)**:
From the equation \( \text{det}(3A) = k \cdot \text{det}(A) \), we can equate:
\[ k \cdot \text{det}(A) = 27 \cdot \text{det}(A) \]
Dividing both sides by \( \text{det}(A) \) (assuming \( \text{det}(A) \neq 0 \)):
\[ k = 27 \]

5. **Conclusion**:
Thus, the value of \( k \) is \( 27 \).

### Final Answer:
The value of \( k \) is \( \boxed{27} \).