If `A = [{:(0,2,0),(0,0,3), (-2,2,0):}] and B =[{:(1,2,3),(3,4,5),(5,-4,0):}],` then the element of third row and column in AB will be

Answer

To find the element of the third row and third column in the product of matrices \( A \) and \( B \), we will follow these steps:

### Step 1: Define the matrices
Given matrices are:
\[ A = \begin{pmatrix} 0 & 2 & 0 \\ 0 & 0 & 3 \\ -2 & 2 & 0 \end{pmatrix} \]
\[ B = \begin{pmatrix} 1 & 2 & 3 \\ 3 & 4 & 5 \\ 5 & -4 & 0 \end{pmatrix} \]

### Step 2: Identify the element to find
We need to find the element at the third row and third column of the product \( AB \).

### Step 3: Calculate the element at the third row and third column
To find the element at position (3,3) in the product \( AB \), we use the formula for matrix multiplication:
\[ (AB)_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj} \]
For our case, \( i = 3 \) and \( j = 3 \):
\[ (AB)_{33} = A_{31}B_{13} + A_{32}B_{23} + A_{33}B_{33} \]

Substituting the values:
- \( A_{31} = -2 \), \( B_{13} = 3 \)
- \( A_{32} = 2 \), \( B_{23} = 5 \)
- \( A_{33} = 0 \), \( B_{33} = 0 \)

Now calculate:
\[ (AB)_{33} = (-2) \cdot (3) + (2) \cdot (5) + (0) \cdot (0) \]
\[ = -6 + 10 + 0 \]
\[ = 4 \]

### Final Answer
The element of the third row and third column in the product \( AB \) is \( 4 \).

---