Given `: M =[(5,-3),(-2, 4)]` find its transpose matrix M'. If possible, find :
(i) M+M`""^(t)` (ii) `M^(t)-M`

Answer

To solve the given problem step by step, we will follow these instructions:

### Step 1: Find the Transpose of Matrix M

Given the matrix:
\[ M = \begin{pmatrix} 5 & -3 \\ -2 & 4 \end{pmatrix} \]

The transpose of a matrix is obtained by swapping its rows with columns. Therefore, the transpose \( M' \) will be:
\[ M' = \begin{pmatrix} 5 & -2 \\ -3 & 4 \end{pmatrix} \]

### Step 2: Calculate \( M + M' \)

Now we will add matrix \( M \) and its transpose \( M' \):
\[ M + M' = \begin{pmatrix} 5 & -3 \\ -2 & 4 \end{pmatrix} + \begin{pmatrix} 5 & -2 \\ -3 & 4 \end{pmatrix} \]

Adding the corresponding elements:
\[ \begin{pmatrix} 5 + 5 & -3 + (-2) \\ -2 + (-3) & 4 + 4 \end{pmatrix} = \begin{pmatrix} 10 & -5 \\ -5 & 8 \end{pmatrix} \]

### Step 3: Calculate \( M' - M \)

Now we will subtract matrix \( M \) from its transpose \( M' \):
\[ M' - M = \begin{pmatrix} 5 & -2 \\ -3 & 4 \end{pmatrix} - \begin{pmatrix} 5 & -3 \\ -2 & 4 \end{pmatrix} \]

Subtracting the corresponding elements:
\[ \begin{pmatrix} 5 - 5 & -2 - (-3) \\ -3 - (-2) & 4 - 4 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \]

### Final Answers

1. The transpose of matrix \( M \) is:
\[ M' = \begin{pmatrix} 5 & -2 \\ -3 & 4 \end{pmatrix} \]

2. The result of \( M + M' \) is:
\[ M + M' = \begin{pmatrix} 10 & -5 \\ -5 & 8 \end{pmatrix} \]

3. The result of \( M' - M \) is:
\[ M' - M = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \]