If `|{:(a,b,c),(m,n,p),(x,y,z):}|=k` then thet value of `|{:(6a,2b,2c),(3m,n,p),(3x,y,z):}|`

Options

A. `k/6`
B. 2k
C. 3k
D. 6k

Answer

To solve the problem, we need to evaluate the determinant given the relationship between the original determinant and the transformed one.

Given:
\[ |{(a,b,c),(m,n,p),(x,y,z)}| = k \]

We need to find:
\[ |{(6a,2b,2c),(3m,n,p),(3x,y,z)}| \]

### Step-by-Step Solution:

1. **Identify the Determinant**: We start with the determinant:
\[ D_1 = |{(a,b,c),(m,n,p),(x,y,z)}| = k \]

2. **Transform the First Row**: The first row of the new determinant is \((6a, 2b, 2c)\). We can factor out constants from this row:
\[ D_2 = |{(6a, 2b, 2c),(3m,n,p),(3x,y,z)}| = 6 \cdot 2 \cdot 2 \cdot |{(a,b,c),(m,n,p),(x,y,z)}| \]
Here, we took \(6\) from the first element of the first row, \(2\) from the second element of the first row, and \(2\) from the third element of the first row.

3. **Transform the Second Row**: Now, we look at the second row \((3m, n, p)\). We can factor out \(3\):
\[ D_2 = 6 \cdot 2 \cdot 2 \cdot 3 \cdot |{(a,b,c),(m,n,p),(x,y,z)}| \]

4. **Combine the Factors**: Now we can combine the constants:
\[ D_2 = 6 \cdot 2 \cdot 2 \cdot 3 \cdot k = 6 \cdot 12 \cdot k = 72k \]

5. **Final Result**: Thus, the value of the determinant is:
\[ |{(6a,2b,2c),(3m,n,p),(3x,y,z)}| = 72k \]

### Final Answer:
\[ |{(6a,2b,2c),(3m,n,p),(3x,y,z)}| = 72k \]