Express `[(3,-4),(1,-1)]` as the sum of symmetric and skew-symmetric matrices.

Answer

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

### Step 1: Define the Matrix
Let \( A = \begin{pmatrix} 3 & -4 \\ 1 & -1 \end{pmatrix} \).

### Step 2: Find the Symmetric and Skew-Symmetric Parts
We can express \( A \) as the sum of a symmetric matrix \( S \) and a skew-symmetric matrix \( K \):
\[ A = S + K \]
Where:
- The symmetric part \( S \) is given by:
\[ S = \frac{1}{2}(A + A^T) \]
- The skew-symmetric part \( K \) is given by:
\[ K = \frac{1}{2}(A - A^T) \]

### Step 3: Calculate the Transpose of A
First, we need to find the transpose of \( A \):
\[ A^T = \begin{pmatrix} 3 & 1 \\ -4 & -1 \end{pmatrix} \]

### Step 4: Calculate the Symmetric Part S
Now, we can calculate \( S \):
\[ S = \frac{1}{2}\left(A + A^T\right) = \frac{1}{2}\left(\begin{pmatrix} 3 & -4 \\ 1 & -1 \end{pmatrix} + \begin{pmatrix} 3 & 1 \\ -4 & -1 \end{pmatrix}\right) \]
Calculating the sum:
\[ A + A^T = \begin{pmatrix} 3 + 3 & -4 + 1 \\ 1 - 4 & -1 - 1 \end{pmatrix} = \begin{pmatrix} 6 & -3 \\ -3 & -2 \end{pmatrix} \]
Now, divide by 2:
\[ S = \frac{1}{2}\begin{pmatrix} 6 & -3 \\ -3 & -2 \end{pmatrix} = \begin{pmatrix} 3 & -\frac{3}{2} \\ -\frac{3}{2} & -1 \end{pmatrix} \]

### Step 5: Calculate the Skew-Symmetric Part K
Now, we can calculate \( K \):
\[ K = \frac{1}{2}\left(A - A^T\right) = \frac{1}{2}\left(\begin{pmatrix} 3 & -4 \\ 1 & -1 \end{pmatrix} - \begin{pmatrix} 3 & 1 \\ -4 & -1 \end{pmatrix}\right) \]
Calculating the difference:
\[ A - A^T = \begin{pmatrix} 3 - 3 & -4 - 1 \\ 1 + 4 & -1 + 1 \end{pmatrix} = \begin{pmatrix} 0 & -5 \\ 5 & 0 \end{pmatrix} \]
Now, divide by 2:
\[ K = \frac{1}{2}\begin{pmatrix} 0 & -5 \\ 5 & 0 \end{pmatrix} = \begin{pmatrix} 0 & -\frac{5}{2} \\ \frac{5}{2} & 0 \end{pmatrix} \]

### Step 6: Write the Final Result
Thus, we can express the original matrix \( A \) as:
\[ A = S + K = \begin{pmatrix} 3 & -\frac{3}{2} \\ -\frac{3}{2} & -1 \end{pmatrix} + \begin{pmatrix} 0 & -\frac{5}{2} \\ \frac{5}{2} & 0 \end{pmatrix} \]

### Summary
The symmetric matrix \( S \) is:
\[ S = \begin{pmatrix} 3 & -\frac{3}{2} \\ -\frac{3}{2} & -1 \end{pmatrix} \]
The skew-symmetric matrix \( K \) is:
\[ K = \begin{pmatrix} 0 & -\frac{5}{2} \\ \frac{5}{2} & 0 \end{pmatrix} \]