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) `[(3,-1,3),(-1,0,2)][(2,-3),(1,0),(3,1)]`.

Answer

Let's solve the given matrix products step by step.

### (i) Compute the product of matrices `[(a,b),(-b,a)]` and `[(a,-b),(b,a)]`.

Let:
\[ A = \begin{pmatrix} a & b \\ -b & a \end{pmatrix} \]
\[ B = \begin{pmatrix} a & -b \\ b & a \end{pmatrix} \]

The product \( AB \) is calculated as follows:

\[ AB = \begin{pmatrix} a & b \\ -b & a \end{pmatrix} \begin{pmatrix} a & -b \\ b & a \end{pmatrix} \]

Calculating each element of the resulting matrix:

1. First row, first column:
\[ a \cdot a + b \cdot b = a^2 + b^2 \]

2. First row, second column:
\[ a \cdot (-b) + b \cdot a = -ab + ab = 0 \]

3. Second row, first column:
\[ -b \cdot a + a \cdot b = -ab + ab = 0 \]

4. Second row, second column:
\[ -b \cdot (-b) + a \cdot a = b^2 + a^2 \]

Thus, the resulting product is:
\[ AB = \begin{pmatrix} a^2 + b^2 & 0 \\ 0 & a^2 + b^2 \end{pmatrix} \]