If `A=[(1,0),(0,0)],B=[(0,1),(0 ,0)]` then AB=………….

Answer

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

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

### Step 1: Write the matrices
We start by writing down the matrices \( A \) and \( B \) clearly.

\[ A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \]

### Step 2: Set up the multiplication
To multiply \( A \) and \( B \), we will calculate each element of the resulting matrix \( AB \).

The resulting matrix \( AB \) will also be a \( 2 \times 2 \) matrix, and its elements are calculated as follows:

\[ AB = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \]

where:
- \( a_{11} = A_{11}B_{11} + A_{12}B_{21} \)
- \( a_{12} = A_{11}B_{12} + A_{12}B_{22} \)
- \( a_{21} = A_{21}B_{11} + A_{22}B_{21} \)
- \( a_{22} = A_{21}B_{12} + A_{22}B_{22} \)

### Step 3: Calculate each element
Now, we will calculate each element of the resulting matrix \( AB \):

1. **Calculate \( a_{11} \)**:
\[ a_{11} = (1)(0) + (0)(0) = 0 + 0 = 0 \]

2. **Calculate \( a_{12} \)**:
\[ a_{12} = (1)(1) + (0)(0) = 1 + 0 = 1 \]

3. **Calculate \( a_{21} \)**:
\[ a_{21} = (0)(0) + (0)(0) = 0 + 0 = 0 \]

4. **Calculate \( a_{22} \)**:
\[ a_{22} = (0)(1) + (0)(0) = 0 + 0 = 0 \]

### Step 4: Write the resulting matrix
Putting it all together, we have:

\[ AB = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \]

### Final Answer
Thus, the product \( AB \) is:

\[ AB = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \]