`(i) if A=[{:(1,0),(0,1):}],B=[{:(0,1),(1,0):}]and C=[{:(1,0),(0,1):}],` then show that `A^(2)=B^(2)=C^(2)=I_(2).` `(ii) if A=[{:(1,0),(1,1):}],B=[{:(2,0),(1,1):}]and C=[{:(-1,2),(3,1):}],` then show that A(B+C)=AB+AC. `(iii) if A=[{:(1,-1),(-1,1):}]and B=[{:(1,1),(1,1):}],`then show that AB is a zero matrix.

Explanation

Let's solve the given questions step by step.

### (i) Show that \( A^2 = B^2 = C^2 = I_2 \)

Given:
- \( A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \)
- \( B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \)
- \( C = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \)

**Step 1: Calculate \( A^2 \)**

\[ A^2 = A \cdot A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

Calculating the product:

\[ A^2 = \begin{pmatrix} 1 \cdot 1 + 0 \cdot 0 & 1 \cdot 0 + 0 \cdot 1 \\ 0 \cdot 1 + 1 \cdot 0 & 0 \cdot 0 + 1 \cdot 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I_2 \]

**Step 2: Calculate \( B^2 \)**

\[ B^2 = B \cdot B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \cdot \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \]

Calculating the product:

\[ B^2 = \begin{pmatrix} 0 \cdot 0 + 1 \cdot 1 & 0 \cdot 1 + 1 \cdot 0 \\ 1 \cdot 0 + 0 \cdot 1 & 1 \cdot 1 + 0 \cdot 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I_2 \]

**Step 3: Calculate \( C^2 \)**

Since \( C = A \), we have:

\[ C^2 = C \cdot C = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I_2 \]

Thus, we have shown that:

\[ A^2 = B^2 = C^2 = I_2 \]