compute the indicated products . `(i) [{:(a,b),(-b,a):}][{:(a,-b),(b,a):}](ii) [{:(1),(2),(3):}][2" "3 " "4 ]` `(iii) [{:(1,-2),(2,3):}][{:(1,2,3),(2,3,1):}]` `(iv)[{:(2,3,4),(3,4,5),(4,5,6):}][{:(1,-3,5),(0,2,4),(3,0,5):}]` (V)`[{:(2,1),(3,2),(-1,1):}][{:(1,0,1),(-1,2,1):}]` (vi) `[{:(3,-1,3),(-1,0,2):}][{:(2,-3),(1,0),(3,1):}]`

Explanation

To solve the given problems, we will compute the indicated products of the matrices step by step.

### (i) Compute the product of matrices \(\begin{pmatrix} a & b \\ -b & a \end{pmatrix}\) and \(\begin{pmatrix} a & -b \\ b & a \end{pmatrix}\)

1. **Matrix Multiplication**:
\[ \begin{pmatrix} a & b \\ -b & a \end{pmatrix} \begin{pmatrix} a & -b \\ b & a \end{pmatrix} = \begin{pmatrix} a \cdot a + b \cdot b & a \cdot (-b) + b \cdot a \\ -b \cdot a + a \cdot b & -b \cdot (-b) + a \cdot a \end{pmatrix} \]

2. **Calculate Each Element**:
- First row, first column: \(a^2 + b^2\)
- First row, second column: \(-ab + ab = 0\)
- Second row, first column: \(-ab + ab = 0\)
- Second row, second column: \(b^2 + a^2\)

3. **Final Result**:
\[ \begin{pmatrix} a^2 + b^2 & 0 \\ 0 & a^2 + b^2 \end{pmatrix} \]