If `A=[{:(,8,6),(,-2,4):}] and B=[{:(,-3,5),(,1,0):}]` then solve for `2 xx 2` matrix X such that
(i) A+X=B
(ii) X-B=A

Answer

To solve the problem step by step, we will find the matrix \( X \) for both equations given in the question.

### Given Matrices:
- \( A = \begin{pmatrix} 8 & 6 \\ -2 & 4 \end{pmatrix} \)
- \( B = \begin{pmatrix} -3 & 5 \\ 1 & 0 \end{pmatrix} \)

### Part (i): Solve for \( A + X = B \)

1. **Rearranging the Equation**:
\[ X = B - A \]

2. **Substituting the Values of \( B \) and \( A \)**:
\[ X = \begin{pmatrix} -3 & 5 \\ 1 & 0 \end{pmatrix} - \begin{pmatrix} 8 & 6 \\ -2 & 4 \end{pmatrix} \]

3. **Performing the Matrix Subtraction**:
\[ X = \begin{pmatrix} -3 - 8 & 5 - 6 \\ 1 - (-2) & 0 - 4 \end{pmatrix} \]
\[ X = \begin{pmatrix} -11 & -1 \\ 3 & -4 \end{pmatrix} \]

### Result for Part (i):
\[ X = \begin{pmatrix} -11 & -1 \\ 3 & -4 \end{pmatrix} \]

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### Part (ii): Solve for \( X - B = A \)

1. **Rearranging the Equation**:
\[ X = A + B \]

2. **Substituting the Values of \( A \) and \( B \)**:
\[ X = \begin{pmatrix} 8 & 6 \\ -2 & 4 \end{pmatrix} + \begin{pmatrix} -3 & 5 \\ 1 & 0 \end{pmatrix} \]

3. **Performing the Matrix Addition**:
\[ X = \begin{pmatrix} 8 + (-3) & 6 + 5 \\ -2 + 1 & 4 + 0 \end{pmatrix} \]
\[ X = \begin{pmatrix} 5 & 11 \\ -1 & 4 \end{pmatrix} \]

### Result for Part (ii):
\[ X = \begin{pmatrix} 5 & 11 \\ -1 & 4 \end{pmatrix} \]

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### Summary of Results:
- For \( A + X = B \), we found:
\[ X = \begin{pmatrix} -11 & -1 \\ 3 & -4 \end{pmatrix} \]
- For \( X - B = A \), we found:
\[ X = \begin{pmatrix} 5 & 11 \\ -1 & 4 \end{pmatrix} \]