If `A=[{:(,1,3),(,2,4):}], B=[{:(,1,2),(,4,3):}] and C=[{:(,4,3),(,1,2):}]`. Find
(i) (AB) C (ii) A (BC)
Is `A(BC)=(AB) C?`

Answer

To solve the problem, we need to find the products of the matrices \( A \), \( B \), and \( C \) in two different orders and check if they are equal.

Given matrices:
- \( A = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix} \)
- \( B = \begin{pmatrix} 1 & 2 \\ 4 & 3 \end{pmatrix} \)
- \( C = \begin{pmatrix} 4 & 3 \\ 1 & 2 \end{pmatrix} \)

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

To find \( AB \), we multiply matrix \( A \) with matrix \( B \):

\[ AB = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 4 & 3 \end{pmatrix} \]

Calculating each element:
- First row, first column: \( 1 \times 1 + 3 \times 4 = 1 + 12 = 13 \)
- First row, second column: \( 1 \times 2 + 3 \times 3 = 2 + 9 = 11 \)
- Second row, first column: \( 2 \times 1 + 4 \times 4 = 2 + 16 = 18 \)
- Second row, second column: \( 2 \times 2 + 4 \times 3 = 4 + 12 = 16 \)

Thus,

\[ AB = \begin{pmatrix} 13 & 11 \\ 18 & 16 \end{pmatrix} \]

### Step 2: Calculate \( (AB)C \)

Now, we will multiply \( AB \) with \( C \):

\[ (AB)C = \begin{pmatrix} 13 & 11 \\ 18 & 16 \end{pmatrix} \begin{pmatrix} 4 & 3 \\ 1 & 2 \end{pmatrix} \]

Calculating each element:
- First row, first column: \( 13 \times 4 + 11 \times 1 = 52 + 11 = 63 \)
- First row, second column: \( 13 \times 3 + 11 \times 2 = 39 + 22 = 61 \)
- Second row, first column: \( 18 \times 4 + 16 \times 1 = 72 + 16 = 88 \)
- Second row, second column: \( 18 \times 3 + 16 \times 2 = 54 + 32 = 86 \)

Thus,

\[ (AB)C = \begin{pmatrix} 63 & 61 \\ 88 & 86 \end{pmatrix} \]

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

Next, we will calculate \( BC \):

\[ BC = \begin{pmatrix} 1 & 2 \\ 4 & 3 \end{pmatrix} \begin{pmatrix} 4 & 3 \\ 1 & 2 \end{pmatrix} \]

Calculating each element:
- First row, first column: \( 1 \times 4 + 2 \times 1 = 4 + 2 = 6 \)
- First row, second column: \( 1 \times 3 + 2 \times 2 = 3 + 4 = 7 \)
- Second row, first column: \( 4 \times 4 + 3 \times 1 = 16 + 3 = 19 \)
- Second row, second column: \( 4 \times 3 + 3 \times 2 = 12 + 6 = 18 \)

Thus,

\[ BC = \begin{pmatrix} 6 & 7 \\ 19 & 18 \end{pmatrix} \]

### Step 4: Calculate \( A(BC) \)

Now, we will multiply \( A \) with \( BC \):

\[ A(BC) = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix} \begin{pmatrix} 6 & 7 \\ 19 & 18 \end{pmatrix} \]

Calculating each element:
- First row, first column: \( 1 \times 6 + 3 \times 19 = 6 + 57 = 63 \)
- First row, second column: \( 1 \times 7 + 3 \times 18 = 7 + 54 = 61 \)
- Second row, first column: \( 2 \times 6 + 4 \times 19 = 12 + 76 = 88 \)
- Second row, second column: \( 2 \times 7 + 4 \times 18 = 14 + 72 = 86 \)

Thus,

\[ A(BC) = \begin{pmatrix} 63 & 61 \\ 88 & 86 \end{pmatrix} \]

### Conclusion

Now we compare \( (AB)C \) and \( A(BC) \):

\[ (AB)C = \begin{pmatrix} 63 & 61 \\ 88 & 86 \end{pmatrix} \]
\[ A(BC) = \begin{pmatrix} 63 & 61 \\ 88 & 86 \end{pmatrix} \]

Since both results are equal, we conclude that:

\[ A(BC) = (AB)C \]