यदि A = $ \left[\begin{array}{r} -2 \\ 4 \\ 5 \end{array}\right]$, B = $\left[\begin{array}{lll} 1 & 3 & -6 \end{array}\right] $ है तो सत्यापित कीजिए (AB)$^{\prime}$ = B$^{\prime}$A$^{\prime} $ है।
EXAMPLE-21
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यहाँ A = $ \left[\begin{array}{r} -2 \\ 4 \\ 5 \end{array}\right]$, B = $\left[\begin{array}{lll} 1 & 3 & -6 \end{array}\right] $
इसलिए AB = $ \left[\begin{array}{r}-2 \\ 4 \\ 5\end{array}\right]$ $\left[\begin{array}{lll}1 & 3 & -6\end{array}\right]$ = $\left[\begin{array}{ccc}-2 & -6 & 12 \\ 4 & 12 & -24 \\ 5 & 15 & -30\end{array}\right]$
अतः (AB)$^{\prime}$ = $\left[\begin{array}{ccc} -2 & 4 & 5 \\ -6 & 12 & 15 \\ 12 & -24 & -30 \end{array}\right] $
अब A$^{\prime}$ = $\left[\begin{array}{lll} -2 & 4 & 5 \end{array}\right]$, B$^{\prime}$ = $ \left[\begin{array}{r} 1 \\ 3 \\ -6 \end{array}\right] $
इसलिए B$^{\prime} $A$^{\prime}$ = $\left[\begin{array}{r}1 \\ 3 \\ -6\end{array}\right]$ $\left[\begin{array}{lll}-2 & 4 & 5\end{array}\right]$ = $ \left[\begin{array}{ccc}-2 & 4 & 5 \\ -6 & 12 & 15 \\ 12 & -24 & -30\end{array}\right]$ = (AB)$^{\prime}$
स्पष्टतया (AB)$^{\prime}$ = B$^{\prime}$A$^{\prime}$
इसलिए AB = $ \left[\begin{array}{r}-2 \\ 4 \\ 5\end{array}\right]$ $\left[\begin{array}{lll}1 & 3 & -6\end{array}\right]$ = $\left[\begin{array}{ccc}-2 & -6 & 12 \\ 4 & 12 & -24 \\ 5 & 15 & -30\end{array}\right]$
अतः (AB)$^{\prime}$ = $\left[\begin{array}{ccc} -2 & 4 & 5 \\ -6 & 12 & 15 \\ 12 & -24 & -30 \end{array}\right] $
अब A$^{\prime}$ = $\left[\begin{array}{lll} -2 & 4 & 5 \end{array}\right]$, B$^{\prime}$ = $ \left[\begin{array}{r} 1 \\ 3 \\ -6 \end{array}\right] $
इसलिए B$^{\prime} $A$^{\prime}$ = $\left[\begin{array}{r}1 \\ 3 \\ -6\end{array}\right]$ $\left[\begin{array}{lll}-2 & 4 & 5\end{array}\right]$ = $ \left[\begin{array}{ccc}-2 & 4 & 5 \\ -6 & 12 & 15 \\ 12 & -24 & -30\end{array}\right]$ = (AB)$^{\prime}$
स्पष्टतया (AB)$^{\prime}$ = B$^{\prime}$A$^{\prime}$
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