If `A=[{:(2,3,-1),(1,4,2):}]andB=[{:(2,3),(4,5),(2,1):}]`, then AB=

Options

A. `[{:(14,20,25),(24,18,16):}]`
B. `[{:(14,24),(20,18),(25,16):}]`
C. `[{:(14,20),(22,25):}]`
D. `4[{:(25,22),(20,14):}]`

Answer

To find the product of the matrices \( A \) and \( B \), we will follow the matrix multiplication rules. Let's denote the matrices as follows:

\[ A = \begin{pmatrix} 2 & 3 & -1 \\ 1 & 4 & 2 \end{pmatrix} \]
\[ B = \begin{pmatrix} 2 & 3 \\ 4 & 5 \\ 2 & 1 \end{pmatrix} \]

### Step 1: Identify the dimensions of the matrices
Matrix \( A \) is a \( 2 \times 3 \) matrix and matrix \( B \) is a \( 3 \times 2 \) matrix. The product \( AB \) will be a \( 2 \times 2 \) matrix.

**Hint:** Remember that the number of columns in the first matrix must equal the number of rows in the second matrix for multiplication to be valid.

### Step 2: Calculate the elements of the resulting matrix \( AB \)

1. **Element at (1,1)**:
\[ AB_{11} = (2 \cdot 2) + (3 \cdot 4) + (-1 \cdot 2) = 4 + 12 - 2 = 14 \]

2. **Element at (1,2)**:
\[ AB_{12} = (2 \cdot 3) + (3 \cdot 5) + (-1 \cdot 1) = 6 + 15 - 1 = 20 \]

3. **Element at (2,1)**:
\[ AB_{21} = (1 \cdot 2) + (4 \cdot 4) + (2 \cdot 2) = 2 + 16 + 4 = 22 \]

4. **Element at (2,2)**:
\[ AB_{22} = (1 \cdot 3) + (4 \cdot 5) + (2 \cdot 1) = 3 + 20 + 2 = 25 \]

### Step 3: Assemble the resulting matrix
Now we can write the resulting matrix \( AB \):
\[ AB = \begin{pmatrix} 14 & 20 \\ 22 & 25 \end{pmatrix} \]

### Final Result
Thus, the product \( AB \) is:
\[ AB = \begin{pmatrix} 14 & 20 \\ 22 & 25 \end{pmatrix} \]