Question

Express the matrix \(A = \begin{bmatrix} 2 & 4 \\ 5 & 6 \end{bmatrix}\) as the sum of a symmetric matrix and a skew symmetric matrix.

Hint:

Use \(P = \frac{1}{2}(A + A^T)\) for the symmetric part and \(Q = \frac{1}{2}(A - A^T)\) for the skew symmetric part.

Answer 1

Step 1: Recall the result:
Every square matrix \(A\) can be written as \(A = P + Q\), where \(P = \dfrac{1}{2}(A + A^T)\) is symmetric and \(Q = \dfrac{1}{2}(A - A^T)\) is skew symmetric.

Step 2: Find the transpose:
\[ A = \begin{bmatrix} 2 & 4 \\ 5 & 6 \end{bmatrix}, \quad A^T = \begin{bmatrix} 2 & 5 \\ 4 & 6 \end{bmatrix} \]

Step 3: Find the symmetric part:
\[ A + A^T = \begin{bmatrix} 4 & 9 \\ 9 & 12 \end{bmatrix} \]
\[ P = \frac{1}{2}(A + A^T) = \begin{bmatrix} 2 & \frac{9}{2} \\ \frac{9}{2} & 6 \end{bmatrix} \]
Here \(P^T = P\), so \(P\) is symmetric.

Step 4: Find the skew symmetric part:
\[ A - A^T = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \]
\[ Q = \frac{1}{2}(A - A^T) = \begin{bmatrix} 0 & -\frac{1}{2} \\ \frac{1}{2} & 0 \end{bmatrix} \]
Here \(Q^T = -Q\), so \(Q\) is skew symmetric.

Step 5: Add them back:
\[ P + Q = \begin{bmatrix} 2 + 0 & \frac{9}{2} - \frac{1}{2} \\ \frac{9}{2} + \frac{1}{2} & 6 + 0 \end{bmatrix} = \begin{bmatrix} 2 & 4 \\ 5 & 6 \end{bmatrix} = A \]

Final Answer:
A is the sum of a symmetric and a skew symmetric matrix: \[ \boxed{\begin{bmatrix} 2 & 4 \\ 5 & 6 \end{bmatrix} = \begin{bmatrix} 2 & \frac{9}{2} \\ \frac{9}{2} & 6 \end{bmatrix} + \begin{bmatrix} 0 & -\frac{1}{2} \\ \frac{1}{2} & 0 \end{bmatrix}} \]