Question

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

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

Answer 1

Answer: (C)

a zero matrix of order $1$

Since $A$ is a Skew-Symmetric matrix

$A^T=-A$

$C$ is a column matrix of order $n\times 1$

So order of $C^T=1\times n$

So order of $C^{T}AC=(1\times n)\times (n\times n)\times (n\times 1)$

$=1\times 1$

Since $C^{T}AC$ has order $=1\times 1$

$\therefore (C^{T}AC{)}^T=C^{T}AC$

Let's assume $C^{T}AC=B$

$(C^{T}AC{)}^T=B^T$

or $C^T{A}^T(C^T{)}^T=B^T$

or $C^T{A}^{T}C=B^T$

or $C^T(-A)C=B^T[\because A^T=-A]$

or $-C^{T}AC=B[\because B^T=B]$

or $-B=B$

or $2B=0$

or $B=0$

$\left(C\right)$ a zero matrix of order $1$