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If i A=[ co...

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If (i) $A = [\begin{matrix} cos α & sin α - sin α & cos α \end{matrix}]$ , then verify that $A^{'} A = I$

(ii) $A = [\begin{matrix} sin α & cos α - cos α & sin α \end{matrix}]$ , then verify that $A^{'} A = I$

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Solution

(i) $A = [\begin{matrix} cos α & sin α - sin α & cos α \end{matrix}]$
$∴ A^{'} = [\begin{matrix} cos α & sin α sin α & cos α \end{matrix}]$
$A^{'} A = [\begin{matrix} cos α & - sin α sin α & cos α \end{matrix}] [\begin{matrix} cos α & sin α - sin α & cos α \end{matrix}]$
$= [\begin{matrix} (cos α) (cos α) + (- sin α) (- sin α) & (cos α) (sin α) + (- sin α) (cos α) (sin α) (cos α) + (cos α) (- sin α) & (sin α) (sin α) + (cos α) (cos α) \end{matrix}]$
$= [\begin{matrix} {c o s}^{2} α + {s i n}^{2} α & sin α cos α - sin α cos α sin α cos α - sin α cos α & {s i n}^{2} α + {c o s}^{2} α \end{matrix}]$
$= [\begin{matrix} 1 & 0 0 & 1 \end{matrix}] = I$
Hence, we have verified that $A^{'} A = I$
(ii) $A = [\begin{matrix} sin α & cos α - cos α & sin α \end{matrix}]$
$∴ A^{'} = [\begin{matrix} sin α & - cos α cos α & sin α \end{matrix}]$
$A^{'} A = [\begin{matrix} sin α & - cos α cos α & sin α \end{matrix}] [\begin{matrix} sin α & cos α - cos α & sin α \end{matrix}]$
$[\begin{matrix} sin α & - cos α cos α & sin α \end{matrix}] [\begin{matrix} sin α & cos α - cos α & sin α \end{matrix}]$
$= [\begin{matrix} (sin α) (sin α) + (- cos α) (- cos α) & (sin α) (cos α) - (- cos α) (sin α) (cos α) (cos α) + (sin α) (- cos α) & (cos α) (cos α) + (sin α) (sin α) \end{matrix}]$
$= [\begin{matrix} {s i n}^{2} α + {c o s}^{2} α & sin α cos α - sin α cos α sin α cos α - sin α cos α & {c o s}^{2} α + {s i n}^{2} α \end{matrix}]$
$= [\begin{matrix} 1 & 0 0 & 1 \end{matrix}] = I$
Hence, we have verified that $A^{'} A = I$

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