If `A = ((1,2),(2,3))and = ((1,0),(1,0))`, then what is the value of determinant of AB ?

Options

A. 0
B. 1
C. 10
D. 20

Answer

To find the value of the determinant of the product of two matrices \( A \) and \( B \), we will follow these steps:

### Step 1: Define the matrices
Let:
\[ A = \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix} \]

### Step 2: Multiply the matrices \( A \) and \( B \)
To find \( AB \), we perform matrix multiplication:
\[ AB = \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix} \]

Calculating the elements of the resulting matrix:
- First row, first column: \( 1 \cdot 1 + 2 \cdot 1 = 1 + 2 = 3 \)
- First row, second column: \( 1 \cdot 0 + 2 \cdot 0 = 0 + 0 = 0 \)
- Second row, first column: \( 2 \cdot 1 + 3 \cdot 1 = 2 + 3 = 5 \)
- Second row, second column: \( 2 \cdot 0 + 3 \cdot 0 = 0 + 0 = 0 \)

Thus, we have:
\[ AB = \begin{pmatrix} 3 & 0 \\ 5 & 0 \end{pmatrix} \]

### Step 3: Calculate the determinant of \( AB \)
The determinant of a \( 2 \times 2 \) matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is calculated as:
\[ \text{det} = ad - bc \]

For our matrix \( AB \):
\[ \text{det}(AB) = (3)(0) - (0)(5) = 0 - 0 = 0 \]

### Conclusion
The value of the determinant of \( AB \) is:
\[ \text{det}(AB) = 0 \]