If matrix `A=[(0,1,-1),(4,-3,4),(3,-3,4)]=B+C`, where B is symmetric matrix and C is skew-symmetric matrix, then find matrices B and C.

Explanation

To find the matrices \( B \) and \( C \) such that \( A = B + C \), where \( B \) is a symmetric matrix and \( C \) is a skew-symmetric matrix, we can follow these steps:

### Step 1: Write down the given matrix \( A \)

Given:
\[ A = \begin{pmatrix} 0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4 \end{pmatrix} \]

### Step 2: Find the transpose of matrix \( A \)

The transpose of matrix \( A \), denoted \( A^T \), is obtained by swapping the rows and columns:
\[ A^T = \begin{pmatrix} 0 & 4 & 3 \\ 1 & -3 & -3 \\ -1 & 4 & 4 \end{pmatrix} \]

### Step 3: Calculate the symmetric matrix \( B \)

The symmetric matrix \( B \) can be calculated using the formula:
\[ B = \frac{1}{2}(A + A^T) \]

Substituting the values of \( A \) and \( A^T \):
\[ B = \frac{1}{2} \left( \begin{pmatrix} 0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4 \end{pmatrix} + \begin{pmatrix} 0 & 4 & 3 \\ 1 & -3 & -3 \\ -1 & 4 & 4 \end{pmatrix} \right) \]

Calculating the sum:
\[ A + A^T = \begin{pmatrix} 0 + 0 & 1 + 4 & -1 + 3 \\ 4 + 1 & -3 - 3 & 4 - 3 \\ 3 - 1 & -3 + 4 & 4 + 4 \end{pmatrix} = \begin{pmatrix} 0 & 5 & 2 \\ 5 & -6 & 1 \\ 2 & 1 & 8 \end{pmatrix} \]

Now, divide by 2:
\[ B = \frac{1}{2} \begin{pmatrix} 0 & 5 & 2 \\ 5 & -6 & 1 \\ 2 & 1 & 8 \end{pmatrix} = \begin{pmatrix} 0 & \frac{5}{2} & 1 \\ \frac{5}{2} & -3 & \frac{1}{2} \\ 1 & \frac{1}{2} & 4 \end{pmatrix} \]

### Step 4: Calculate the skew-symmetric matrix \( C \)

The skew-symmetric matrix \( C \) can be calculated using the formula:
\[ C = \frac{1}{2}(A - A^T) \]

Substituting the values of \( A \) and \( A^T \):
\[ C = \frac{1}{2} \left( \begin{pmatrix} 0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4 \end{pmatrix} - \begin{pmatrix} 0 & 4 & 3 \\ 1 & -3 & -3 \\ -1 & 4 & 4 \end{pmatrix} \right) \]

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

Now, divide by 2:
\[ C = \frac{1}{2} \begin{pmatrix} 0 & -3 & -4 \\ 3 & 0 & 7 \\ 4 & -7 & 0 \end{pmatrix} = \begin{pmatrix} 0 & -\frac{3}{2} & -2 \\ \frac{3}{2} & 0 & \frac{7}{2} \\ 2 & -\frac{7}{2} & 0 \end{pmatrix} \]

### Final Result

Thus, the matrices \( B \) and \( C \) are:
\[ B = \begin{pmatrix} 0 & \frac{5}{2} & 1 \\ \frac{5}{2} & -3 & \frac{1}{2} \\ 1 & \frac{1}{2} & 4 \end{pmatrix} \]
\[ C = \begin{pmatrix} 0 & -\frac{3}{2} & -2 \\ \frac{3}{2} & 0 & \frac{7}{2} \\ 2 & -\frac{7}{2} & 0 \end{pmatrix} \]