express the following matrices as a sum of symmetric and skew symmetric matrices :
`(i) [{:(3,-7),(4,2):}] `

Explanation

To express the given matrix as a sum of symmetric and skew-symmetric matrices, we can follow these steps:

### Given Matrix:
Let \( A = \begin{pmatrix} 3 & -7 \\ 4 & 2 \end{pmatrix} \)

### Step 1: Find the Transpose of Matrix A
The transpose of a matrix is obtained by swapping its rows and columns.

\[ A^T = \begin{pmatrix} 3 & 4 \\ -7 & 2 \end{pmatrix} \]

### Step 2: Calculate the Symmetric Matrix
The symmetric part of the matrix can be calculated using the formula:

\[ S = \frac{1}{2}(A + A^T) \]

Substituting the values of \( A \) and \( A^T \):

\[ S = \frac{1}{2} \left( \begin{pmatrix} 3 & -7 \\ 4 & 2 \end{pmatrix} + \begin{pmatrix} 3 & 4 \\ -7 & 2 \end{pmatrix} \right) \]

Calculating the sum:

\[ = \frac{1}{2} \begin{pmatrix} 3 + 3 & -7 + 4 \\ 4 - 7 & 2 + 2 \end{pmatrix} = \frac{1}{2} \begin{pmatrix} 6 & -3 \\ -3 & 4 \end{pmatrix} \]

Now, multiplying by \( \frac{1}{2} \):

\[ S = \begin{pmatrix} 3 & -\frac{3}{2} \\ -\frac{3}{2} & 2 \end{pmatrix} \]

### Step 3: Calculate the Skew-Symmetric Matrix
The skew-symmetric part of the matrix can be calculated using the formula:

\[ K = \frac{1}{2}(A - A^T) \]

Substituting the values of \( A \) and \( A^T \):

\[ K = \frac{1}{2} \left( \begin{pmatrix} 3 & -7 \\ 4 & 2 \end{pmatrix} - \begin{pmatrix} 3 & 4 \\ -7 & 2 \end{pmatrix} \right) \]

Calculating the difference:

\[ = \frac{1}{2} \begin{pmatrix} 3 - 3 & -7 - 4 \\ 4 + 7 & 2 - 2 \end{pmatrix} = \frac{1}{2} \begin{pmatrix} 0 & -11 \\ 11 & 0 \end{pmatrix} \]

Now, multiplying by \( \frac{1}{2} \):

\[ K = \begin{pmatrix} 0 & -\frac{11}{2} \\ \frac{11}{2} & 0 \end{pmatrix} \]

### Step 4: Express A as the Sum of S and K
Now we can express the original matrix \( A \) as the sum of the symmetric matrix \( S \) and the skew-symmetric matrix \( K \):

\[ A = S + K \]

Thus, we have:

\[ A = \begin{pmatrix} 3 & -\frac{3}{2} \\ -\frac{3}{2} & 2 \end{pmatrix} + \begin{pmatrix} 0 & -\frac{11}{2} \\ \frac{11}{2} & 0 \end{pmatrix} \]

### Final Result
The original matrix \( A \) can be expressed as:

\[ A = \begin{pmatrix} 3 & -\frac{3}{2} \\ -\frac{3}{2} & 2 \end{pmatrix} + \begin{pmatrix} 0 & -\frac{11}{2} \\ \frac{11}{2} & 0 \end{pmatrix} \]