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If $A=\left(\begin{array}{c}\alpha-1 \\ 0 \\ 0\end{array}\right), B=\left(\begin{array}{c}\alpha+1 \\ 0 \\ 0\end{array}\right)$ be two matrices, then $A B^T$ is a non-zero matrix for $|\alpha|$ not equal to
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Let $A=\left(\begin{array}{c}\alpha-1 \\ 0 \\ 0\end{array}\right), B=\left(\begin{array}{c}\alpha+1 \\ 0 \\ 0\end{array}\right)$ be two matrices.
$$
\begin{aligned}
& A B^T=\left(\begin{array}{c}
\alpha-1 \\
0 \\
0
\end{array}\right)(\alpha+1 \quad 0 \quad 0) \\
& =\left(\begin{array}{ccc}
\alpha^2-1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}\right) \\
&
\end{aligned}
$$
$$
\begin{aligned}
& A B^T=\left(\begin{array}{c}
\alpha-1 \\
0 \\
0
\end{array}\right)(\alpha+1 \quad 0 \quad 0) \\
& =\left(\begin{array}{ccc}
\alpha^2-1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}\right) \\
&
\end{aligned}
$$
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