Question

If $A=\begin{pmatrix}\alpha-1\\ 0\\ 0\end{pmatrix}, B=\begin{pmatrix}\alpha+1\\ 0\\ 0\end{pmatrix}$ be two matrices, then $AB^T$ is a non-zero matrix for $|\alpha|$ not equal to

• A. $2$
• B. $0$
• C. $1$
• D. $3$

Answer 1

Answer: (C)

Let $A=\begin{pmatrix}\alpha-1\\ 0\\ 0\end{pmatrix}, B=\begin{pmatrix}\alpha+1\\ 0\\ 0\end{pmatrix} $
be two matrices.
$AB^{T}=\begin{pmatrix}\alpha -1\\ 0\\ 0\end{pmatrix}\left(\alpha+1\quad 0\quad 0\right)$
$=\begin{pmatrix}\alpha^{2}-1&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}$
Thus, $AB^T$ is non-zero matrix for $\left|\alpha\right| \ne 1$