Question

Divide the matrix A in the sum of symmetric and skew-symmetric

A = \(\rm \begin{bmatrix} 2 & -4 & 3\\ 3 & 1 & -2\\ 1& -3 & 5 \end{bmatrix}\).

Which of the following is that skew-symmetric matrix?

1. \(\rm \begin{bmatrix} 0 & -3.5 & 1\\ -3.5& 0 & 0.5\\ 1& 0.5 & 0 \end{bmatrix}\)
2. \(\rm \begin{bmatrix} 2 & -0.5 & 2\\ 0.5& 1 & -2.5\\ -2& 2.5 & 5 \end{bmatrix}\)
3. \(\rm \begin{bmatrix} 0 & -3.5 & 1\\ 3.5& 0 & 0.5\\ -1& -0.5 & 0 \end{bmatrix}\)
4. \(\rm \begin{bmatrix} 2 & -0.5 & 2\\ -0.5& 1 & -2.5\\ 2& -2.5 & 5 \end{bmatrix}\)

Answer 1

Correct Answer - Option 3 : \(\rm \begin{bmatrix} 0 & -3.5 & 1\\ 3.5& 0 & 0.5\\ -1& -0.5 & 0 \end{bmatrix}\)

Concept:

A matrix X can be written as a sum of symmetric and skew-symmetric matrix which are 

P(symmetric) = \(\rm 1\over2\)(X + X^T)

Q(skew-symmetric) = \(\rm 1\over2\)(X - XT)

Calculation:

A = \(\rm \begin{bmatrix} 2 & -4 & 3\\ 3 & 1 & -2\\ 1& -3 & 5 \end{bmatrix}\)

A^T = \(\rm \begin{bmatrix} 2 & 3 & 1\\ -4& 1 & -3\\ 3& -2 & 5 \end{bmatrix}\)

P(symmetric matrix) = \(\rm 1\over2\)(A + AT)

⇒ P = \(\rm {1\over2}\left(\begin{bmatrix} 2 & -4 & 3\\ 3 & 1 & -2\\ 1& -3 & 5 \end{bmatrix}+ \begin{bmatrix} 2 & 3 & 1\\ -4& 1 & -3\\ 3& -2 & 5 \end{bmatrix}\right)\)

⇒ P = \(\rm \begin{bmatrix} 2 & -0.5 & 2\\ -0.5& 1 & -2.5\\ 2& -2.5 & 5 \end{bmatrix}\)

Q(skew-symmetric) = \(\rm 1\over2\)(A - AT)

⇒ Q = \(\rm {1\over2}\left(\begin{bmatrix} 2 & -4 & 3\\ 3 & 1 & -2\\ 1& -3 & 5 \end{bmatrix}- \begin{bmatrix} 2 & 3 & 1\\ -4& 1 & -3\\ 3& -2 & 5 \end{bmatrix}\right)\) 

 

⇒ Q = \(\rm \begin{bmatrix} 0 & -3.5 & 1\\ 3.5& 0 & 0.5\\ -1& -0.5 & 0 \end{bmatrix}\)