If `Delta = |{:(1,2,3),(2,3,5),(3,6,12):}|and Delta' = |{:(4,8,15),(3,6,12),(2,3,5):}|` , then

Options

A. `Delta' = 2 Delta`
B. `Delta' = - 2 Delta`
C. `Delta' = Delta`
D. `Delta' = - Delta`

Answer

To solve the problem, we need to calculate the values of the determinants \( \Delta \) and \( \Delta' \) and then find the relationship between them.

### Step 1: Calculate \( \Delta \)

Given:
\[ \Delta = \begin{vmatrix} 1 & 2 & 3 \\ 2 & 3 & 5 \\ 3 & 6 & 12 \end{vmatrix} \]

We will expand this determinant using the first row.

\[ \Delta = 1 \cdot \begin{vmatrix} 3 & 5 \\ 6 & 12 \end{vmatrix} - 2 \cdot \begin{vmatrix} 2 & 5 \\ 3 & 12 \end{vmatrix} + 3 \cdot \begin{vmatrix} 2 & 3 \\ 3 & 6 \end{vmatrix} \]

### Step 2: Calculate the 2x2 determinants

1. For \( \begin{vmatrix} 3 & 5 \\ 6 & 12 \end{vmatrix} \):
\[ = (3 \cdot 12) - (5 \cdot 6) = 36 - 30 = 6 \]

2. For \( \begin{vmatrix} 2 & 5 \\ 3 & 12 \end{vmatrix} \):
\[ = (2 \cdot 12) - (5 \cdot 3) = 24 - 15 = 9 \]

3. For \( \begin{vmatrix} 2 & 3 \\ 3 & 6 \end{vmatrix} \):
\[ = (2 \cdot 6) - (3 \cdot 3) = 12 - 9 = 3 \]

### Step 3: Substitute back into the determinant calculation

Now substituting these values back into the equation for \( \Delta \):

\[ \Delta = 1 \cdot 6 - 2 \cdot 9 + 3 \cdot 3 \]
\[ = 6 - 18 + 9 \]
\[ = 6 - 18 + 9 = -3 \]

### Step 4: Calculate \( \Delta' \)

Now we calculate \( \Delta' \):

Given:
\[ \Delta' = \begin{vmatrix} 4 & 8 & 15 \\ 3 & 6 & 12 \\ 2 & 3 & 5 \end{vmatrix} \]

We will expand this determinant using the first row.

\[ \Delta' = 4 \cdot \begin{vmatrix} 6 & 12 \\ 3 & 5 \end{vmatrix} - 8 \cdot \begin{vmatrix} 3 & 12 \\ 2 & 5 \end{vmatrix} + 15 \cdot \begin{vmatrix} 3 & 6 \\ 2 & 3 \end{vmatrix} \]

### Step 5: Calculate the 2x2 determinants for \( \Delta' \)

1. For \( \begin{vmatrix} 6 & 12 \\ 3 & 5 \end{vmatrix} \):
\[ = (6 \cdot 5) - (12 \cdot 3) = 30 - 36 = -6 \]

2. For \( \begin{vmatrix} 3 & 12 \\ 2 & 5 \end{vmatrix} \):
\[ = (3 \cdot 5) - (12 \cdot 2) = 15 - 24 = -9 \]

3. For \( \begin{vmatrix} 3 & 6 \\ 2 & 3 \end{vmatrix} \):
\[ = (3 \cdot 3) - (6 \cdot 2) = 9 - 12 = -3 \]

### Step 6: Substitute back into the determinant calculation for \( \Delta' \)

Now substituting these values back into the equation for \( \Delta' \):

\[ \Delta' = 4 \cdot (-6) - 8 \cdot (-9) + 15 \cdot (-3) \]
\[ = -24 + 72 - 45 \]
\[ = -24 + 72 - 45 = 3 \]

### Step 7: Find the relationship between \( \Delta \) and \( \Delta' \)

We have:
\[ \Delta = -3 \quad \text{and} \quad \Delta' = 3 \]

Thus, we can see that:
\[ \Delta = -\Delta' \]

### Final Answer

The relationship is:
\[ \Delta = -\Delta' \]