यदि $A = \left[\begin{array}{rrr}0 & 6 & 7 \\ -6 & 0 & 8 \\ 7 & -8 & 0\end{array}\right], B = \left[\begin{array}{lll}0 & 1 & 1 \\ 1 & 0 & 2 \\ 1 & 2 & 0\end{array}\right], C = \left[\begin{array}{r}2 \\ -2 \\ 3\end{array}\right]$
तो $AC, BC $ तथा $(A + B) \ C$ का परिकलन कीजिए। यह भी सत्यापित कीजिए कि $(A + B) C = AC + BC$
तो $AC, BC $ तथा $(A + B) \ C$ का परिकलन कीजिए। यह भी सत्यापित कीजिए कि $(A + B) C = AC + BC$
EXAMPLE-17
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$A + B = \left[\begin{array}{rrr} 0 & 7 & 8 \\ -5 & 0 & 10 \\ 8 & -6 & 0 \end{array}\right]$
अतएव $, (A + B) C = \left[\begin{array}{rrr}0 & 7 & 8 \\ -5 & 0 & 10 \\ 8 & -6 & 0\end{array}\right] \left[\begin{array}{r}2 \\ -2 \\ 3\end{array}\right] = \left[\begin{array}{r}0-14+24 \\ -10+0+30 \\ 16+12+0\end{array}\right] = \left[\begin{array}{r}10 \\ 20 \\ 28\end{array}\right]$
इसके अतिरिक्त
$AC = \left[\begin{array}{ccc} 0 & 6 & 7 \\ -6 & 0 & 8 \\ 7 & -8 & 0 \end{array}\right] \left[\begin{array}{r} 2 \\ -2 \\ 3 \end{array}\right] = \left[\begin{array}{r} 0-12+21 \\ -12+0+24 \\ 14+16+0 \end{array}\right]= \left[\begin{array}{c} 9 \\ 12 \\ 30 \end{array}\right] $
और $ BC = \left[\begin{array}{lll} 0 & 1 & 1 \\ 1 & 0 & 2 \\ 1 & 2 & 0 \end{array}\right] \left[\begin{array}{r} 2 \\ -2 \\ 3 \end{array}\right] = \left[\begin{array}{l} 0-2+3 \\ 2+0+6 \\ 2-4+0 \end{array}\right] = \left[\begin{array}{c} 1 \\ 8 \\ -2 \end{array}\right]$
इसलिए $AC + BC = \left[\begin{array}{c} 9 \\ 12 \\ 30 \end{array}\right] + \left[\begin{array}{r} 1 \\ 8 \\ -2 \end{array}\right] = \left[\begin{array}{r} 10 \\ 20 \\ 28 \end{array}\right]$
स्पष्टतया $(A + B) C = AC + BC$
अतएव $, (A + B) C = \left[\begin{array}{rrr}0 & 7 & 8 \\ -5 & 0 & 10 \\ 8 & -6 & 0\end{array}\right] \left[\begin{array}{r}2 \\ -2 \\ 3\end{array}\right] = \left[\begin{array}{r}0-14+24 \\ -10+0+30 \\ 16+12+0\end{array}\right] = \left[\begin{array}{r}10 \\ 20 \\ 28\end{array}\right]$
इसके अतिरिक्त
$AC = \left[\begin{array}{ccc} 0 & 6 & 7 \\ -6 & 0 & 8 \\ 7 & -8 & 0 \end{array}\right] \left[\begin{array}{r} 2 \\ -2 \\ 3 \end{array}\right] = \left[\begin{array}{r} 0-12+21 \\ -12+0+24 \\ 14+16+0 \end{array}\right]= \left[\begin{array}{c} 9 \\ 12 \\ 30 \end{array}\right] $
और $ BC = \left[\begin{array}{lll} 0 & 1 & 1 \\ 1 & 0 & 2 \\ 1 & 2 & 0 \end{array}\right] \left[\begin{array}{r} 2 \\ -2 \\ 3 \end{array}\right] = \left[\begin{array}{l} 0-2+3 \\ 2+0+6 \\ 2-4+0 \end{array}\right] = \left[\begin{array}{c} 1 \\ 8 \\ -2 \end{array}\right]$
इसलिए $AC + BC = \left[\begin{array}{c} 9 \\ 12 \\ 30 \end{array}\right] + \left[\begin{array}{r} 1 \\ 8 \\ -2 \end{array}\right] = \left[\begin{array}{r} 10 \\ 20 \\ 28 \end{array}\right]$
स्पष्टतया $(A + B) C = AC + BC$
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