If `A=[(3,2),(2,-1)]` and `B=[(1,-2),(3,0)]` , then the value of AB is :

Options

A. `[(9,6),(1,4)]`
B. `[(-9,-6),(-1,-4)]`
C. `[(9,-6),(-1,-4)]`
D. None of these

Answer

To find the product of the matrices \( A \) and \( B \), we will follow the matrix multiplication rules step by step.

Given:
\[ A = \begin{pmatrix} 3 & 2 \\ 2 & -1 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & -2 \\ 3 & 0 \end{pmatrix} \]

### Step 1: Verify the dimensions
The dimensions of matrix \( A \) are \( 2 \times 2 \) and the dimensions of matrix \( B \) are also \( 2 \times 2 \). Since the number of columns in \( A \) (which is 2) is equal to the number of rows in \( B \) (which is also 2), the product \( AB \) is defined.

**Hint:** Always check the dimensions of the matrices before performing multiplication.

### Step 2: Calculate the elements of the product matrix \( AB \)
The product \( AB \) will also be a \( 2 \times 2 \) matrix. We will calculate each element of the resulting matrix.

1. **Element at position (1,1)**:
\[ AB_{11} = (3 \times 1) + (2 \times 3) = 3 + 6 = 9 \]

2. **Element at position (1,2)**:
\[ AB_{12} = (3 \times -2) + (2 \times 0) = -6 + 0 = -6 \]

3. **Element at position (2,1)**:
\[ AB_{21} = (2 \times 1) + (-1 \times 3) = 2 - 3 = -1 \]

4. **Element at position (2,2)**:
\[ AB_{22} = (2 \times -2) + (-1 \times 0) = -4 + 0 = -4 \]

### Step 3: Form the resulting matrix
Combining all the calculated elements, we get:
\[ AB = \begin{pmatrix} 9 & -6 \\ -1 & -4 \end{pmatrix} \]

### Final Answer
The value of \( AB \) is:
\[ \begin{pmatrix} 9 & -6 \\ -1 & -4 \end{pmatrix} \]

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