Question

If c is a skew symmetric matrix of order n×n and x is a column matrix of order n×1 then prove that xtcx=0

Answer 1

Given: c is a skew symmetric matrix of order n×n and x is a column matrix of order n×1.

To find: Prove that x^T * c * x = 0

Solution:

• As we have given that c is skew symmetric matrix of order nxn, so let the value of n be 3 and matrix be:

                     $\left[\begin{array}{ccc}0&2&3\\-2&0&4\\-3&-4&0\end{array}\right]$

• Also let x be matrix as:

                     $\left[\begin{array}{c}1&2&3\\\end{array}\right]$

• Now x^T will be:

                     $\left[\begin{array}{ccc}1&2&3\\\end{array}\right]$

• Now we have to evaluate x^T * c * x
• So,

                   =   $\left[\begin{array}{ccc}1&2&3\\\end{array}\right]$$\left[\begin{array}{ccc}0&2&3\\-2&0&4\\-3&-4&0\end{array}\right]$$\left[\begin{array}{ccc}1\\2\\3\end{array}\right]$

• So, we get:

                   =   $\left[\begin{array}{ccc}-13&-10&11\end{array}\right]$$\left[\begin{array}{ccc}1\\2\\3\end{array}\right]$

                   =   [0]

• Hence proved.

Answer:

             So we have proved that x^T * c * x = 0.