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

Options

A. `[(4,2,6),(-1,2,3),(6,4,9)]`
B. `[(4,1,13),(8,-1,5),(-10,11,-25)]`
C. `[(-2,1,3),(6,9,11),(-1,-7,25)]`
D. `[(2,9,13),(4,7,-11),(-10,11,-20)]`

Answer

To evaluate the product of matrices \( A \) and \( B \), we will follow the matrix multiplication rules. The matrices are defined as follows:

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

### Step 1: Determine the dimensions of the matrices
Matrix \( A \) is a \( 3 \times 3 \) matrix and matrix \( B \) is also a \( 3 \times 3 \) matrix. Since the number of columns in \( A \) matches the number of rows in \( B \), we can multiply them.

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

### Step 2: Calculate the elements of the resulting matrix \( AB \)
The resulting matrix \( AB \) will also be a \( 3 \times 3 \) matrix. Each element \( c_{ij} \) of the resulting matrix \( C = AB \) is calculated as follows:

\[ c_{ij} = \sum_{k=1}^{n} a_{ik} \cdot b_{kj} \]

where \( n \) is the number of columns in \( A \) (or rows in \( B \)).

### Step 3: Calculate each element of the resulting matrix \( C \)

1. **Element \( c_{11} \)**:
\[ c_{11} = 1 \cdot 2 + 2 \cdot (-4) + 5 \cdot 2 = 2 - 8 + 10 = 4 \]

2. **Element \( c_{12} \)**:
\[ c_{12} = 1 \cdot 2 + 2 \cdot 2 + 5 \cdot (-1) = 2 + 4 - 5 = 1 \]

3. **Element \( c_{13} \)**:
\[ c_{13} = 1 \cdot (-4) + 2 \cdot (-4) + 5 \cdot 5 = -4 - 8 + 25 = 13 \]

4. **Element \( c_{21} \)**:
\[ c_{21} = 1 \cdot 2 + (-1) \cdot (-4) + 1 \cdot 2 = 2 + 4 + 2 = 8 \]

5. **Element \( c_{22} \)**:
\[ c_{22} = 1 \cdot 2 + (-1) \cdot 2 + 1 \cdot (-1) = 2 - 2 - 1 = -1 \]

6. **Element \( c_{23} \)**:
\[ c_{23} = 1 \cdot (-4) + (-1) \cdot (-4) + 1 \cdot 5 = -4 + 4 + 5 = 5 \]

7. **Element \( c_{31} \)**:
\[ c_{31} = 2 \cdot 2 + 3 \cdot (-4) + (-1) \cdot 2 = 4 - 12 - 2 = -10 \]

8. **Element \( c_{32} \)**:
\[ c_{32} = 2 \cdot 2 + 3 \cdot 2 + (-1) \cdot (-1) = 4 + 6 + 1 = 11 \]

9. **Element \( c_{33} \)**:
\[ c_{33} = 2 \cdot (-4) + 3 \cdot (-4) + (-1) \cdot 5 = -8 - 12 - 5 = -25 \]

### Step 4: Write the resulting matrix \( C \)
Combining all the calculated elements, we get:

\[ C = AB = \begin{pmatrix} 4 & 1 & 13 \\ 8 & -1 & 5 \\ -10 & 11 & -25 \end{pmatrix} \]

### Final Result
The product of matrices \( A \) and \( B \) is:

\[ AB = \begin{pmatrix} 4 & 1 & 13 \\ 8 & -1 & 5 \\ -10 & 11 & -25 \end{pmatrix} \]