If every element of a third order determinant of value `Detlta` is multiplied by 5, then the value of new determinant, is

Options

A. `Delta`
B. `5 Delta`
C. `25 Delta`
D. `125 Delta`

Explanation

To solve the problem, we need to determine the value of a new determinant when every element of a third-order determinant with value \( \Delta \) is multiplied by 5.

### Step-by-Step Solution:

1. **Understanding the Determinant**:
Let’s denote the original third-order determinant as \( \Delta \). The determinant can be represented as:
\[ \Delta = \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} \]

2. **Multiplying Each Element by 5**:
When we multiply every element of the determinant by 5, the new determinant becomes:
\[ 5 \cdot \Delta = \begin{vmatrix} 5a & 5b & 5c \\ 5d & 5e & 5f \\ 5g & 5h & 5i \end{vmatrix} \]

3. **Factoring Out the Constant**:
According to the properties of determinants, if each element of a row (or column) of a determinant is multiplied by a constant \( k \), the value of the determinant is multiplied by \( k \). For a third-order determinant, if we multiply each of the three rows by 5, the determinant is multiplied by \( 5^3 \):
\[ \Delta' = 5^3 \cdot \Delta \]

4. **Calculating \( 5^3 \)**:
Now, we calculate \( 5^3 \):
\[ 5^3 = 125 \]

5. **Final Value of the New Determinant**:
Therefore, the value of the new determinant \( \Delta' \) is:
\[ \Delta' = 125 \cdot \Delta \]

### Conclusion:
The value of the new determinant after multiplying every element of the original determinant by 5 is \( 125 \Delta \).