If A = ( * 20 c - 1 0 0 ), , , ,B = ( * 20 c + 1 0 0 ) be two mat… - Vidyadip

Question

If $A = \left( {\begin{array}{*{20}{c}}
{\alpha - 1}\\
0\\
0
\end{array}} \right),\,\,\,B = \left( {\begin{array}{*{20}{c}}
{\alpha + 1}\\
0\\
0
\end{array}} \right)$ be two matrices, then $AB^T$ is a non-zero matrix for $\left| \alpha \right|$ not equal to

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Answer

c
Let $A = \left( {\begin{array}{*{20}{c}}
{\alpha - 1}\\
0\\
0
\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}
{\alpha + 1}\\
0\\
0
\end{array}} \right)$

be two matrices.

$A{B^T} = \left( {\begin{array}{*{20}{c}}
{\alpha - 1}\\
0\\
0
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
{\alpha + 1}&0&0
\end{array}} \right)$

$ = \left( {\begin{array}{*{20}{c}}
{{\alpha ^2} - 1}&0&0\\
0&0&0\\
0&0&0
\end{array}} \right)$

Thus, $A{B^T}$ is non-zero matrix for $\left| \alpha \right| \ne 1$

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