If reach element of a `3xx3` matrix is multiplied by 3, then the determine of the newly formed matrix is

Options

A. `3|A|`
B. `9|A|`
C. `27|A|`
D. `|A|^(3)`

Answer

To find the determinant of a newly formed matrix when each element of a \(3 \times 3\) matrix is multiplied by 3, we can follow these steps:

### Step-by-Step Solution:

1. **Let the original matrix be \(A\)**:
\[ A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \]

2. **Multiply each element of the matrix \(A\) by 3**:
\[ 3A = \begin{pmatrix} 3a_{11} & 3a_{12} & 3a_{13} \\ 3a_{21} & 3a_{22} & 3a_{23} \\ 3a_{31} & 3a_{32} & 3a_{33} \end{pmatrix} \]

3. **Use the property of determinants**:
The determinant of a matrix that has been multiplied by a scalar \(k\) can be expressed as:
\[ \text{det}(kA) = k^n \cdot \text{det}(A) \]
where \(n\) is the order of the matrix. In this case, since we have a \(3 \times 3\) matrix, \(n = 3\).

4. **Substituting the values**:
Here, \(k = 3\) and \(n = 3\):
\[ \text{det}(3A) = 3^3 \cdot \text{det}(A) = 27 \cdot \text{det}(A) \]

5. **Conclusion**:
Therefore, the determinant of the newly formed matrix \(3A\) is:
\[ \text{det}(3A) = 27 \cdot \text{det}(A) \]