Express the matrix `[(2,4,-6),(7,3,5),(1,-2,4)]` as a sum of symmetric and a skew - symmetric matrix.

Answer

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

1. The symmetric part \( P \) is given by:
\[ P = \frac{1}{2}(A + A^T) \]

2. The skew-symmetric part \( Q \) is given by:
\[ Q = \frac{1}{2}(A - A^T) \]

### Step 1: Find the transpose of matrix \( A \)
The transpose of matrix \( A \), denoted as \( A^T \), is obtained by swapping rows with columns:
\[ A^T = \begin{pmatrix} 2 & 7 & 1 \\ 4 & 3 & -2 \\ -6 & 5 & 4 \end{pmatrix} \]

### Step 2: Calculate the symmetric matrix \( P \)
Now, we can calculate \( P \) using the formula:
\[ P = \frac{1}{2}(A + A^T) \]
Substituting the values of \( A \) and \( A^T \):
\[ P = \frac{1}{2} \left( \begin{pmatrix} 2 & 4 & -6 \\ 7 & 3 & 5 \\ 1 & -2 & 4 \end{pmatrix} + \begin{pmatrix} 2 & 7 & 1 \\ 4 & 3 & -2 \\ -6 & 5 & 4 \end{pmatrix} \right) \]

Calculating the sum:
\[ A + A^T = \begin{pmatrix} 2 + 2 & 4 + 7 & -6 + 1 \\ 7 + 4 & 3 + 3 & 5 - 2 \\ 1 - 6 & -2 + 5 & 4 + 4 \end{pmatrix} = \begin{pmatrix} 4 & 11 & -5 \\ 11 & 6 & 3 \\ -5 & 3 & 8 \end{pmatrix} \]

Now, divide by 2:
\[ P = \frac{1}{2} \begin{pmatrix} 4 & 11 & -5 \\ 11 & 6 & 3 \\ -5 & 3 & 8 \end{pmatrix} = \begin{pmatrix} 2 & \frac{11}{2} & -\frac{5}{2} \\ \frac{11}{2} & 3 & \frac{3}{2} \\ -\frac{5}{2} & \frac{3}{2} & 4 \end{pmatrix} \]

### Step 3: Calculate the skew-symmetric matrix \( Q \)
Now, we can calculate \( Q \) using the formula:
\[ Q = \frac{1}{2}(A - A^T) \]
Substituting the values of \( A \) and \( A^T \):
\[ Q = \frac{1}{2} \left( \begin{pmatrix} 2 & 4 & -6 \\ 7 & 3 & 5 \\ 1 & -2 & 4 \end{pmatrix} - \begin{pmatrix} 2 & 7 & 1 \\ 4 & 3 & -2 \\ -6 & 5 & 4 \end{pmatrix} \right) \]

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

Now, divide by 2:
\[ Q = \frac{1}{2} \begin{pmatrix} 0 & -3 & -7 \\ 3 & 0 & 7 \\ 7 & -7 & 0 \end{pmatrix} = \begin{pmatrix} 0 & -\frac{3}{2} & -\frac{7}{2} \\ \frac{3}{2} & 0 & \frac{7}{2} \\ \frac{7}{2} & -\frac{7}{2} & 0 \end{pmatrix} \]

### Final Result
Thus, we can express the matrix \( A \) as:
\[ A = P + Q \]
Where:
\[ P = \begin{pmatrix} 2 & \frac{11}{2} & -\frac{5}{2} \\ \frac{11}{2} & 3 & \frac{3}{2} \\ -\frac{5}{2} & \frac{3}{2} & 4 \end{pmatrix} \]
And:
\[ Q = \begin{pmatrix} 0 & -\frac{3}{2} & -\frac{7}{2} \\ \frac{3}{2} & 0 & \frac{7}{2} \\ \frac{7}{2} & -\frac{7}{2} & 0 \end{pmatrix} \]