If `A=[(2,-3,1),(-2,3,4)]` and `B=[(2,5),(3,1),(4,2)]`, then Find AB

Answer

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

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

### Step 1: Check the dimensions of the matrices

Matrix \( A \) has dimensions \( 2 \times 3 \) (2 rows and 3 columns), and matrix \( B \) has dimensions \( 3 \times 2 \) (3 rows and 2 columns).

Since the number of columns in \( A \) (3) is equal to the number of rows in \( B \) (3), we can multiply these matrices. The resulting matrix \( AB \) will have dimensions \( 2 \times 2 \).

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

The element in the resulting matrix \( AB \) at position \( (i, j) \) is calculated by taking the dot product of the \( i \)-th row of matrix \( A \) and the \( j \)-th column of matrix \( B \).

1. **Calculate \( (1,1) \) element:**
\[ AB[1,1] = (2 \times 2) + (-3 \times 3) + (1 \times 4) = 4 - 9 + 4 = -1 \]

2. **Calculate \( (1,2) \) element:**
\[ AB[1,2] = (2 \times 5) + (-3 \times 1) + (1 \times 2) = 10 - 3 + 2 = 9 \]

3. **Calculate \( (2,1) \) element:**
\[ AB[2,1] = (-2 \times 2) + (3 \times 3) + (4 \times 4) = -4 + 9 + 16 = 21 \]

4. **Calculate \( (2,2) \) element:**
\[ AB[2,2] = (-2 \times 5) + (3 \times 1) + (4 \times 2) = -10 + 3 + 8 = 1 \]

### Step 3: Write the resulting matrix

Combining all the calculated elements, we get:
\[ AB = \begin{pmatrix} -1 & 9 \\ 21 & 1 \end{pmatrix} \]

### Final Result
Thus, the product of matrices \( A \) and \( B \) is:
\[ AB = \begin{pmatrix} -1 & 9 \\ 21 & 1 \end{pmatrix} \]