If `Delta_1=|(1,0),(a,b)|` and `Delta_2=|(1,0),(c,d)|`, then `Delta_2Delta_1` is equal to :

Options

A. ac
B. bd
C. (b-a)(d-c)
D. None of these

Answer

To solve the problem, we need to calculate the determinants \(\Delta_1\) and \(\Delta_2\) and then find the product \(\Delta_2 \Delta_1\).

### Step 1: Calculate \(\Delta_1\)

Given:
\[ \Delta_1 = \begin{vmatrix} 1 & 0 \\ a & b \end{vmatrix} \]

The determinant of a 2x2 matrix \(\begin{vmatrix} p & q \\ r & s \end{vmatrix}\) is calculated as:
\[ \text{det} = ps - qr \]

Applying this to \(\Delta_1\):
\[ \Delta_1 = (1 \cdot b) - (0 \cdot a) = b - 0 = b \]

### Step 2: Calculate \(\Delta_2\)

Given:
\[ \Delta_2 = \begin{vmatrix} 1 & 0 \\ c & d \end{vmatrix} \]

Using the same determinant formula:
\[ \Delta_2 = (1 \cdot d) - (0 \cdot c) = d - 0 = d \]

### Step 3: Calculate \(\Delta_2 \Delta_1\)

Now we find the product:
\[ \Delta_2 \Delta_1 = d \cdot b = bd \]

### Final Answer

Thus, \(\Delta_2 \Delta_1 = bd\).

### Options

The correct option is:
- Option 2: \(bd\)

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