Express the matrix A = [(3,2,5)(4,1,3)(0,6,7)] as sum of two matrices such that one is symmetric and the other is skew- symmetric.

Question

Express the matrix A = \( \begin{bmatrix} 3 &2 & 5 \\[0.3em] 4 &1 & 3\\[0.3em] 0 & 6& 7 \end{bmatrix}\)as sum of two matrices such that one is symmetric and the other is skew-symmetric.

A = [(3,2,5)(4,1,3)(0,6,7)]

Answer 1

Given A = \( \begin{bmatrix} 3 &2&5 \\[0.3em] 4 & 1 & 3 \\[0.3em] 0 & 6& 7 \end{bmatrix}\), to express as sum of symmetric matrix P and skew symmetric matrix Q.

A = P + Q

Where P = \(\frac{1}{2}\)(A + A') and Q = \(\frac{1}{2}\)(A - A')

First we will find A’

A' = \( \begin{bmatrix} 3 &4&0 \\[0.3em] 2& 1 & 6\\[0.3em] 5 & 3& 7 \end{bmatrix}\)

Now using above mentioned formulas

P = \(\frac{1}{2}\)(A + A')

Q = \(\frac{1}{2}\)(A - A')

Now A = P + Q

\(\Rightarrow\) \( \begin{bmatrix} 3 &2&5 \\[0.3em] 4 & 1 & 3 \\[0.3em] 0 & 6& 7 \end{bmatrix}\)