If `Delta` is the value of the determinant
`|(a_(1), b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))|`
then what is the value of the following determinant?
`|(pa_(1), b_(1), qc_(1)),(pa_(2), b_(2), qc_(2)),(pa_(3), b_(3),qc_(3))|`
`(p ne 0 or 1, q ne 0 or 1)`

Options

A. `p Delta`
B. `q Delta`
C. `(p+q)Delta`
D. `pq Delta`

Answer

To solve the given problem, we need to evaluate the determinant:

\[ D = \begin{vmatrix} pa_1 & b_1 & qc_1 \\ pa_2 & b_2 & qc_2 \\ pa_3 & b_3 & qc_3 \end{vmatrix} \]

### Step 1: Factor out constants from the columns
We can factor out the constants \(p\) and \(q\) from the first and third columns respectively.

\[ D = p \cdot q \cdot \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} \]

### Step 2: Identify the remaining determinant
The remaining determinant is the original determinant given in the problem, which we denote as \(\Delta\):

\[ \Delta = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} \]

### Step 3: Substitute back into the expression for \(D\)
Now we substitute \(\Delta\) back into the expression for \(D\):

\[ D = p \cdot q \cdot \Delta \]

### Final Result
Thus, the value of the determinant is:

\[ D = pq \Delta \]