If each element of a determinant of third order with value A is multiplied by 3, then the value of newly formed determinant is

Options

A. 3A
B. 9A
C. 27A
D. none of these

Answer

To solve the problem, we need to understand how the value of a determinant changes when we multiply each element of the determinant by a constant.

### Step-by-Step Solution:

1. **Understanding the Determinant**:
Let the original determinant of a 3x3 matrix be represented as:
\[ D = \begin{vmatrix} A & B & C \\ D & E & F \\ G & H & I \end{vmatrix} \]
where the value of this determinant is given as \( A \).

2. **Multiplying Each Element by 3**:
When each element of the determinant is multiplied by 3, the new matrix becomes:
\[ D' = \begin{vmatrix} 3A & 3B & 3C \\ 3D & 3E & 3F \\ 3G & 3H & 3I \end{vmatrix} \]

3. **Effect on the Determinant**:
The property of determinants states that if each element of a determinant of order \( n \) is multiplied by a constant \( k \), then the value of the determinant is multiplied by \( k^n \). In this case, since we have a 3x3 matrix (order 3), and we are multiplying each element by 3:
\[ \text{New Determinant Value} = 3^3 \cdot D \]

4. **Calculating the New Value**:
Since \( D \) (the original determinant value) is \( A \):
\[ \text{New Determinant Value} = 27 \cdot A \]

5. **Final Result**:
Therefore, the value of the newly formed determinant after multiplying each element by 3 is:
\[ 27A \]