Let `A=[{:(,5,4),(,3,-2):}], B=[{:(,-3,0),(,1,4):}] and C=[{:(,1,-3),(,0,2):}],` find :
(i) A+B and B+A (ii) (A+B)+C and A+(B+C)
(iii) Is A+B=B+A? (iv) Is (A+B) +C=A+(B+C)?

Answer

To solve the problem step by step, we will perform the matrix additions as specified in the question.

### Given Matrices:
- \( A = \begin{pmatrix} 5 & 4 \\ 3 & -2 \end{pmatrix} \)
- \( B = \begin{pmatrix} -3 & 0 \\ 1 & 4 \end{pmatrix} \)
- \( C = \begin{pmatrix} 1 & -3 \\ 0 & 2 \end{pmatrix} \)

### (i) Finding \( A + B \) and \( B + A \)

**Step 1: Calculate \( A + B \)**

\[ A + B = \begin{pmatrix} 5 & 4 \\ 3 & -2 \end{pmatrix} + \begin{pmatrix} -3 & 0 \\ 1 & 4 \end{pmatrix} \]

Now, we add the corresponding elements:

\[ = \begin{pmatrix} 5 + (-3) & 4 + 0 \\ 3 + 1 & -2 + 4 \end{pmatrix} \]

Calculating each element:

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

**Step 2: Calculate \( B + A \)**

\[ B + A = \begin{pmatrix} -3 & 0 \\ 1 & 4 \end{pmatrix} + \begin{pmatrix} 5 & 4 \\ 3 & -2 \end{pmatrix} \]

Adding the corresponding elements:

\[ = \begin{pmatrix} -3 + 5 & 0 + 4 \\ 1 + 3 & 4 + (-2) \end{pmatrix} \]

Calculating each element:

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

### (ii) Finding \( (A + B) + C \) and \( A + (B + C) \)

**Step 3: Calculate \( (A + B) + C \)**

We already found \( A + B = \begin{pmatrix} 2 & 4 \\ 4 & 2 \end{pmatrix} \).

Now, we add \( C \):

\[ (A + B) + C = \begin{pmatrix} 2 & 4 \\ 4 & 2 \end{pmatrix} + \begin{pmatrix} 1 & -3 \\ 0 & 2 \end{pmatrix} \]

Adding the corresponding elements:

\[ = \begin{pmatrix} 2 + 1 & 4 + (-3) \\ 4 + 0 & 2 + 2 \end{pmatrix} \]

Calculating each element:

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

**Step 4: Calculate \( A + (B + C) \)**

First, we need to calculate \( B + C \):

\[ B + C = \begin{pmatrix} -3 & 0 \\ 1 & 4 \end{pmatrix} + \begin{pmatrix} 1 & -3 \\ 0 & 2 \end{pmatrix} \]

Adding the corresponding elements:

\[ = \begin{pmatrix} -3 + 1 & 0 + (-3) \\ 1 + 0 & 4 + 2 \end{pmatrix} \]

Calculating each element:

\[ = \begin{pmatrix} -2 & -3 \\ 1 & 6 \end{pmatrix} \]

Now, we add \( A \):

\[ A + (B + C) = \begin{pmatrix} 5 & 4 \\ 3 & -2 \end{pmatrix} + \begin{pmatrix} -2 & -3 \\ 1 & 6 \end{pmatrix} \]

Adding the corresponding elements:

\[ = \begin{pmatrix} 5 + (-2) & 4 + (-3) \\ 3 + 1 & -2 + 6 \end{pmatrix} \]

Calculating each element:

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

### (iii) Is \( A + B = B + A \)?

From our calculations:

\[ A + B = \begin{pmatrix} 2 & 4 \\ 4 & 2 \end{pmatrix} \]
\[ B + A = \begin{pmatrix} 2 & 4 \\ 4 & 2 \end{pmatrix} \]

Since both results are equal, we conclude:

**Yes, \( A + B = B + A \)**.

### (iv) Is \( (A + B) + C = A + (B + C) \)?

From our calculations:

\[ (A + B) + C = \begin{pmatrix} 3 & 1 \\ 4 & 4 \end{pmatrix} \]
\[ A + (B + C) = \begin{pmatrix} 3 & 1 \\ 4 & 4 \end{pmatrix} \]

Since both results are equal, we conclude:

**Yes, \( (A + B) + C = A + (B + C) \)**.

### Summary of Results:
1. \( A + B = \begin{pmatrix} 2 & 4 \\ 4 & 2 \end{pmatrix} \)
2. \( B + A = \begin{pmatrix} 2 & 4 \\ 4 & 2 \end{pmatrix} \)
3. \( (A + B) + C = \begin{pmatrix} 3 & 1 \\ 4 & 4 \end{pmatrix} \)
4. \( A + (B + C) = \begin{pmatrix} 3 & 1 \\ 4 & 4 \end{pmatrix} \)
5. \( A + B = B + A \) (Yes)
6. \( (A + B) + C = A + (B + C) \) (Yes)