Question

If $A$ is a skew symmetric matrix of order $3$ , $B$ is a $3\times 1$ column matrix and $C=B^{T}AB$ , then which of the following is false?

• A. $C$ is singular
• B. $C$ is non singular
• C. $C$ is a symmetric matrix
• D. $C$ is a skew symmetric matrix

Answer 1

Answer: (B)

Given, $C=B^{T}AB$
$C^{T}=\left(B^{T} A B\right)^{T}$
$\Rightarrow C^{T}=B^{T}A^{T}\left(B^{T}\right)^{T}$
$\Rightarrow C^{T}=-B^{T}AB=-C\Rightarrow C$ is skew symmetric
$\because $ order of $C$ is $1\times 1\Rightarrow C=\left[0\right]$
$\Rightarrow C$ is singular and symmetric also.