If `A=[(3,5),(2,0)]` and `B=[(1,17),(0,-10)]` then `|AB|` is equal to

Options

A. 80
B. 100
C. -110
D. 92

Explanation

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

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

Given the matrices:

\[ A = \begin{pmatrix} 3 & 5 \\ 2 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 17 \\ 0 & -10 \end{pmatrix} \]

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

The determinant of a \( 2 \times 2 \) matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is calculated using the formula:

\[ |A| = ad - bc \]

For matrix \( A \):

- \( a = 3 \)
- \( b = 5 \)
- \( c = 2 \)
- \( d = 0 \)

Now substituting these values into the formula:

\[ |A| = (3)(0) - (5)(2) = 0 - 10 = -10 \]

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

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

For matrix \( B \):

- \( a = 1 \)
- \( b = 17 \)
- \( c = 0 \)
- \( d = -10 \)

Now substituting these values into the formula:

\[ |B| = (1)(-10) - (17)(0) = -10 - 0 = -10 \]

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

Using the property of determinants:

\[ |AB| = |A| \cdot |B| = (-10) \cdot (-10) = 100 \]

Thus, the determinant of the product \( AB \) is:

\[ |AB| = 100 \]

### Final Answer:

\[ |AB| = 100 \]