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  4. If A is a skew symmetric matrix of order n and C is a column matrix of order n × 1, then CTAC is:

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Q. If A is a skew symmetric matrix of order n and C is a column matrix of order n $\times $ 1, then $C^TAC$ is:

Matrices

Solution:

Given : A be a skew symmetric matrix of order n.
i.e. $A^T = - (A)_{n \times n}$
C be the column matrix of order n $\times$ 1 then $C^TAC$ is of order 1 $\times$ 1.
Let $C^TAC = \alpha$, Consider $(C^T AC)^T = C^T \, A^TC = C^T(-A) \, C = - C^T AC = - \alpha$
$ \Rightarrow \, \alpha = - \alpha \, \Rightarrow \, \alpha = 0$
Thus, $C^T \, AC$ is zero matrix of order 1.