Compute : `[(1,0),(0,1)][(2,-1),(1,2)]`

Answer

To compute the product of the matrices \(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) and \(\begin{pmatrix} 2 & -1 \\ 1 & 2 \end{pmatrix}\), we will follow the rules of matrix multiplication.

### Step-by-Step Solution:

1. **Identify the matrices**:
- Let \( A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \) (the identity matrix)
- Let \( B = \begin{pmatrix} 2 & -1 \\ 1 & 2 \end{pmatrix} \)

2. **Set up the multiplication**:
- The product \( AB \) will be a \( 2 \times 2 \) matrix, and we will denote it as \( C = AB \).

3. **Calculate each element of the resulting matrix**:
- The element in the first row, first column of \( C \):
\[ C_{11} = (1 \times 2) + (0 \times 1) = 2 + 0 = 2 \]
- The element in the first row, second column of \( C \):
\[ C_{12} = (1 \times -1) + (0 \times 2) = -1 + 0 = -1 \]
- The element in the second row, first column of \( C \):
\[ C_{21} = (0 \times 2) + (1 \times 1) = 0 + 1 = 1 \]
- The element in the second row, second column of \( C \):
\[ C_{22} = (0 \times -1) + (1 \times 2) = 0 + 2 = 2 \]

4. **Combine the results**:
- Thus, the resulting matrix \( C \) is:
\[ C = \begin{pmatrix} 2 & -1 \\ 1 & 2 \end{pmatrix} \]

### Final Result:
The product of the matrices is:
\[ \begin{pmatrix} 2 & -1 \\ 1 & 2 \end{pmatrix} \]