Express the following as the sum of symmetric and skew - symmetric matrices :
`[(1,2,3),(3,4,5),(5,6,7)]`

Answer

To express the matrix \( A = \begin{pmatrix} 1 & 2 & 3 \\ 3 & 4 & 5 \\ 5 & 6 & 7 \end{pmatrix} \) as the sum of a symmetric matrix and a skew-symmetric matrix, we can follow these steps:

### Step 1: Find the transpose of the matrix \( A \)
First, we need to calculate the transpose of matrix \( A \).

\[ A^T = \begin{pmatrix} 1 & 3 & 5 \\ 2 & 4 & 6 \\ 3 & 5 & 7 \end{pmatrix} \]

**Hint:** The transpose of a matrix is obtained by swapping its rows and columns.

### Step 2: Calculate the symmetric matrix \( P \)
The symmetric part \( P \) of the matrix \( A \) can be calculated using the formula:

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

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

\[ P = \frac{1}{2} \left( \begin{pmatrix} 1 & 2 & 3 \\ 3 & 4 & 5 \\ 5 & 6 & 7 \end{pmatrix} + \begin{pmatrix} 1 & 3 & 5 \\ 2 & 4 & 6 \\ 3 & 5 & 7 \end{pmatrix} \right) \]

Calculating the sum inside the parentheses:

\[ A + A^T = \begin{pmatrix} 1+1 & 2+3 & 3+5 \\ 3+2 & 4+4 & 5+6 \\ 5+3 & 6+5 & 7+7 \end{pmatrix} = \begin{pmatrix} 2 & 5 & 8 \\ 5 & 8 & 11 \\ 8 & 11 & 14 \end{pmatrix} \]

Now, divide by 2:

\[ P = \frac{1}{2} \begin{pmatrix} 2 & 5 & 8 \\ 5 & 8 & 11 \\ 8 & 11 & 14 \end{pmatrix} = \begin{pmatrix} 1 & \frac{5}{2} & 4 \\ \frac{5}{2} & 4 & \frac{11}{2} \\ 4 & \frac{11}{2} & 7 \end{pmatrix} \]

**Hint:** The symmetric matrix \( P \) is obtained by averaging the original matrix and its transpose.

### Step 3: Calculate the skew-symmetric matrix \( Q \)
The skew-symmetric part \( Q \) of the matrix \( A \) can be calculated using the formula:

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

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

\[ Q = \frac{1}{2} \left( \begin{pmatrix} 1 & 2 & 3 \\ 3 & 4 & 5 \\ 5 & 6 & 7 \end{pmatrix} - \begin{pmatrix} 1 & 3 & 5 \\ 2 & 4 & 6 \\ 3 & 5 & 7 \end{pmatrix} \right) \]

Calculating the difference inside the parentheses:

\[ A - A^T = \begin{pmatrix} 1-1 & 2-3 & 3-5 \\ 3-2 & 4-4 & 5-6 \\ 5-3 & 6-5 & 7-7 \end{pmatrix} = \begin{pmatrix} 0 & -1 & -2 \\ 1 & 0 & -1 \\ 2 & 1 & 0 \end{pmatrix} \]

Now, divide by 2:

\[ Q = \frac{1}{2} \begin{pmatrix} 0 & -1 & -2 \\ 1 & 0 & -1 \\ 2 & 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & -\frac{1}{2} & -1 \\ \frac{1}{2} & 0 & -\frac{1}{2} \\ 1 & \frac{1}{2} & 0 \end{pmatrix} \]

**Hint:** The skew-symmetric matrix \( Q \) is obtained by averaging the difference between the original matrix and its transpose.

### Step 4: Verify the results
Now, we can verify that \( A = P + Q \):

\[ P + Q = \begin{pmatrix} 1 & \frac{5}{2} & 4 \\ \frac{5}{2} & 4 & \frac{11}{2} \\ 4 & \frac{11}{2} & 7 \end{pmatrix} + \begin{pmatrix} 0 & -\frac{1}{2} & -1 \\ \frac{1}{2} & 0 & -\frac{1}{2} \\ 1 & \frac{1}{2} & 0 \end{pmatrix} \]

Calculating the sum:

\[ = \begin{pmatrix} 1 + 0 & \frac{5}{2} - \frac{1}{2} & 4 - 1 \\ \frac{5}{2} + \frac{1}{2} & 4 + 0 & \frac{11}{2} - \frac{1}{2} \\ 4 + 1 & \frac{11}{2} + \frac{1}{2} & 7 + 0 \end{pmatrix} = \begin{pmatrix} 1 & 2 & 3 \\ 3 & 4 & 5 \\ 5 & 6 & 7 \end{pmatrix} \]

Thus, we have confirmed that \( A = P + Q \).

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

\[ A = P + Q \]
where
\[ P = \begin{pmatrix} 1 & \frac{5}{2} & 4 \\ \frac{5}{2} & 4 & \frac{11}{2} \\ 4 & \frac{11}{2} & 7 \end{pmatrix} \text{ (symmetric)} \]
and
\[ Q = \begin{pmatrix} 0 & -\frac{1}{2} & -1 \\ \frac{1}{2} & 0 & -\frac{1}{2} \\ 1 & \frac{1}{2} & 0 \end{pmatrix} \text{ (skew-symmetric)} \]