If `A=[{:(,1,2),(,2,1):}] and B=[{:(,2,1),(,1,2):}]` find :
(i) A(BA) (ii) (AB) B.

Answer

To solve the problem, we need to find the following:

1. \( A(BA) \)
2. \( (AB)B \)

Given matrices:
\[ A = \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} \]
\[ B = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} \]

### Step 1: Calculate \( BA \)

To find \( BA \), we multiply matrix \( B \) by matrix \( A \):

\[ BA = B \cdot A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} \cdot \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} \]

Calculating the elements of \( BA \):

- First row, first column: \( 2 \cdot 1 + 1 \cdot 2 = 2 + 2 = 4 \)
- First row, second column: \( 2 \cdot 2 + 1 \cdot 1 = 4 + 1 = 5 \)
- Second row, first column: \( 1 \cdot 1 + 2 \cdot 2 = 1 + 4 = 5 \)
- Second row, second column: \( 1 \cdot 2 + 2 \cdot 1 = 2 + 2 = 4 \)

Thus,
\[ BA = \begin{pmatrix} 4 & 5 \\ 5 & 4 \end{pmatrix} \]

### Step 2: Calculate \( A(BA) \)

Now we need to calculate \( A(BA) \):

\[ A(BA) = A \cdot (BA) = \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} \cdot \begin{pmatrix} 4 & 5 \\ 5 & 4 \end{pmatrix} \]

Calculating the elements of \( A(BA) \):

- First row, first column: \( 1 \cdot 4 + 2 \cdot 5 = 4 + 10 = 14 \)
- First row, second column: \( 1 \cdot 5 + 2 \cdot 4 = 5 + 8 = 13 \)
- Second row, first column: \( 2 \cdot 4 + 1 \cdot 5 = 8 + 5 = 13 \)
- Second row, second column: \( 2 \cdot 5 + 1 \cdot 4 = 10 + 4 = 14 \)

Thus,
\[ A(BA) = \begin{pmatrix} 14 & 13 \\ 13 & 14 \end{pmatrix} \]

### Step 3: Calculate \( AB \)

Next, we calculate \( AB \):

\[ AB = A \cdot B = \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} \cdot \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} \]

Calculating the elements of \( AB \):

- First row, first column: \( 1 \cdot 2 + 2 \cdot 1 = 2 + 2 = 4 \)
- First row, second column: \( 1 \cdot 1 + 2 \cdot 2 = 1 + 4 = 5 \)
- Second row, first column: \( 2 \cdot 2 + 1 \cdot 1 = 4 + 1 = 5 \)
- Second row, second column: \( 2 \cdot 1 + 1 \cdot 2 = 2 + 2 = 4 \)

Thus,
\[ AB = \begin{pmatrix} 4 & 5 \\ 5 & 4 \end{pmatrix} \]

### Step 4: Calculate \( (AB)B \)

Now we need to calculate \( (AB)B \):

\[ (AB)B = (AB) \cdot B = \begin{pmatrix} 4 & 5 \\ 5 & 4 \end{pmatrix} \cdot \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} \]

Calculating the elements of \( (AB)B \):

- First row, first column: \( 4 \cdot 2 + 5 \cdot 1 = 8 + 5 = 13 \)
- First row, second column: \( 4 \cdot 1 + 5 \cdot 2 = 4 + 10 = 14 \)
- Second row, first column: \( 5 \cdot 2 + 4 \cdot 1 = 10 + 4 = 14 \)
- Second row, second column: \( 5 \cdot 1 + 4 \cdot 2 = 5 + 8 = 13 \)

Thus,
\[ (AB)B = \begin{pmatrix} 13 & 14 \\ 14 & 13 \end{pmatrix} \]

### Final Results

1. \( A(BA) = \begin{pmatrix} 14 & 13 \\ 13 & 14 \end{pmatrix} \)
2. \( (AB)B = \begin{pmatrix} 13 & 14 \\ 14 & 13 \end{pmatrix} \)