Question

If $A$ is a skew-symmetric matrix of order $n$ and $C$ is a column matrix of the order of $n\times 1$, Then $C^{T}AC$ is

• A. an identity matrix of order $n$
• B. an identity matrix of order $1$
• C. a zero matrix of order $1$
• D. None of these

Answer 1

The correct option is C

a zero matrix of order $1$

Explanation for the correct option:

Find the value of $C^{T}AC$:

Given, $A$ is a $n\times n$ matrix, $c$ is $n\times 1$; $c^T=1\times n$order.

As we know,

$\begin{array}{rcl}{\left(A^{T}AC\right)}^T& =& C^T{A}^T{\left(C^T\right)}^T\\ & =& C^T{A}^{T}C\\ & =& C^T\left(-A\right)C\end{array}$

Let $C^T\left(-A\right)C=K$

$\Rightarrow$ $-C^{T}AC=-K$

$\therefore K=-K$

$\Rightarrow$$2K=0$

$\mathbf{\therefore }\mathit{K}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}$

Thus, $C^{T}AC$ is a zero matrix of order $\mathbf{1}$

Hence, Option ‘C’ is Correct.