Given `A=[{:(,4,1),(,2,3):}] and B=[{:(,1,0),(,-2,01):}]`, Find
(i) A-B (ii) `A^2`
(iii) AB (iv) `A^2-AB+2B`

Answer

To solve the given problem step by step, we will perform the following calculations:

Given matrices:
\[ A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix} \]
\[ B = \begin{pmatrix} 1 & 0 \\ -2 & 1 \end{pmatrix} \]

### (i) Find \( A - B \)

To subtract matrices \( A \) and \( B \), we subtract the corresponding elements:

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

Calculating each element:
\[ = \begin{pmatrix} 3 & 1 \\ 4 & 2 \end{pmatrix} \]

### (ii) Find \( A^2 \)

To find \( A^2 \), we multiply matrix \( A \) by itself:

\[ A^2 = A \times A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix} \times \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix} \]

Calculating the elements:
- First row, first column: \( 4 \times 4 + 1 \times 2 = 16 + 2 = 18 \)
- First row, second column: \( 4 \times 1 + 1 \times 3 = 4 + 3 = 7 \)
- Second row, first column: \( 2 \times 4 + 3 \times 2 = 8 + 6 = 14 \)
- Second row, second column: \( 2 \times 1 + 3 \times 3 = 2 + 9 = 11 \)

Thus,
\[ A^2 = \begin{pmatrix} 18 & 7 \\ 14 & 11 \end{pmatrix} \]

### (iii) Find \( AB \)

To find the product \( AB \):

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

Calculating the elements:
- First row, first column: \( 4 \times 1 + 1 \times (-2) = 4 - 2 = 2 \)
- First row, second column: \( 4 \times 0 + 1 \times 1 = 0 + 1 = 1 \)
- Second row, first column: \( 2 \times 1 + 3 \times (-2) = 2 - 6 = -4 \)
- Second row, second column: \( 2 \times 0 + 3 \times 1 = 0 + 3 = 3 \)

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

### (iv) Find \( A^2 - AB + 2B \)

First, we need to calculate \( 2B \):

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

Now, we can calculate \( A^2 - AB + 2B \):

\[ A^2 - AB + 2B = \begin{pmatrix} 18 & 7 \\ 14 & 11 \end{pmatrix} - \begin{pmatrix} 2 & 1 \\ -4 & 3 \end{pmatrix} + \begin{pmatrix} 2 & 0 \\ -4 & 2 \end{pmatrix} \]

Calculating \( A^2 - AB \):

\[ A^2 - AB = \begin{pmatrix} 18 - 2 & 7 - 1 \\ 14 - (-4) & 11 - 3 \end{pmatrix} = \begin{pmatrix} 16 & 6 \\ 18 & 8 \end{pmatrix} \]

Now adding \( 2B \):

\[ A^2 - AB + 2B = \begin{pmatrix} 16 & 6 \\ 18 & 8 \end{pmatrix} + \begin{pmatrix} 2 & 0 \\ -4 & 2 \end{pmatrix} = \begin{pmatrix} 16 + 2 & 6 + 0 \\ 18 - 4 & 8 + 2 \end{pmatrix} = \begin{pmatrix} 18 & 6 \\ 14 & 10 \end{pmatrix} \]

### Final Answers:
(i) \( A - B = \begin{pmatrix} 3 & 1 \\ 4 & 2 \end{pmatrix} \)

(ii) \( A^2 = \begin{pmatrix} 18 & 7 \\ 14 & 11 \end{pmatrix} \)

(iii) \( AB = \begin{pmatrix} 2 & 1 \\ -4 & 3 \end{pmatrix} \)

(iv) \( A^2 - AB + 2B = \begin{pmatrix} 18 & 6 \\ 14 & 10 \end{pmatrix} \)