Let A`=[{:(200,50),(10,2):}]` and `B=[{:(50,40),(2,3):}]`, then `|AB|` is equal to

Options

A. 460
B. 2000
C. 3000
D. `-7000`

Answer

To find the determinant of the product of two matrices \( A \) and \( B \), we can use the property that states:

\[ |AB| = |A| \cdot |B| \]

Given the matrices:

\[ A = \begin{pmatrix} 200 & 50 \\ 10 & 2 \end{pmatrix} \]

\[ B = \begin{pmatrix} 50 & 40 \\ 2 & 3 \end{pmatrix} \]

We will first calculate the determinant of matrix \( A \) and then the determinant of matrix \( B \).

### Step 1: Calculate the determinant of \( A \)

The determinant of a \( 2 \times 2 \) matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:

\[ |A| = ad - bc \]

For matrix \( A \):

- \( a = 200 \)
- \( b = 50 \)
- \( c = 10 \)
- \( d = 2 \)

Calculating the determinant:

\[ |A| = (200 \cdot 2) - (50 \cdot 10) \]

\[ |A| = 400 - 500 = -100 \]

### Step 2: Calculate the determinant of \( B \)

Using the same formula for matrix \( B \):

For matrix \( B \):

- \( a = 50 \)
- \( b = 40 \)
- \( c = 2 \)
- \( d = 3 \)

Calculating the determinant:

\[ |B| = (50 \cdot 3) - (40 \cdot 2) \]

\[ |B| = 150 - 80 = 70 \]

### Step 3: Calculate the determinant of \( AB \)

Now, using the property of determinants:

\[ |AB| = |A| \cdot |B| \]

Substituting the values we found:

\[ |AB| = (-100) \cdot 70 \]

\[ |AB| = -7000 \]

Thus, the final answer is:

\[ |AB| = -7000 \]