यदि A = $ \left[\begin{array}{ccc}1 & 1 & -1 \\ 2 & 0 & 3 \\ 3 & -1 & 2\end{array}\right]$, B = $\left[\begin{array}{cc}1 & 3 \\ 0 & 2 \\ -1 & 4\end{array}\right]$ तथा C = $\left[\begin{array}{cccc}1 & 2 & 3 & -4 \\ 2 & 0 & -2 & 1\end{array}\right]$ तो A(BC) तथा (AB)C ज्ञात कीजिए और दिखलाइए कि (AB)C = A(BC) है।
EXAMPLE-16
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यहाँ AB = $\left[\begin{array}{ccr} 1 & 1 & -1 \\ 2 & 0 & 3 \\ 3 & -1 & 2 \end{array}\right]$$ \left[\begin{array}{rr} 1 & 3 \\ 0 & 2 \\ -1 & 4 \end{array}\right]$ = $ \left[\begin{array}{cc} 1+0+1 & 3+2-4 \\ 2+0-3 & 6+0+12 \\ 3+0-2 & 9-2+8 \end{array}\right]$= $\left[\begin{array}{cc} 2 & 1 \\ -1 & 18 \\ 1 & 15 \end{array}\right]$
(AB) (C) = $\left[\begin{array}{rc} 2 & 1 \\ -1 & 18 \\ 1 & 15 \end{array}\right]$ $\left[\begin{array}{rrrr} 1 & 2 & 3 & -4 \\ 2 & 0 & -2 & 1 \end{array}\right]$ = $\left[\begin{array}{crcc} 2+2 & 4+0 & 6-2 & -8+1 \\ -1+36 & -2+0 & -3-36 & 4+18 \\ 1+30 & 2+0 & 3-30 & -4+15 \end{array}\right]$
= $\left[\begin{array}{lrcc} 4 & 4 & 4 & -7 \\ 35 & -2 & -39 & 22 \\ 31 & 2 & -27 & 11 \end{array}\right]$
अब BC = $\left[\begin{array}{rr}1 & 3 \\ 0 & 2 \\ -1 & 4\end{array}\right]$ $\left[\begin{array}{rrrr} 1 & 2 & 3 & -4 \\ 2 & 0 & -2 & 1 \end{array}\right]$ = $\left[\begin{array}{rrrr} 1+6 & 2+0 & 3-6 & -4+3 \\ 0+4 & 0+0 & 0-4 & 0+2 \\ -1+8 & -2+0 & -3-8 & 4+4 \end{array}\right]$
= $\left[\begin{array}{cccc} 7 & 2 & -3 & -1 \\ 4 & 0 & -4 & 2 \\ 7 & -2 & -11 & 8 \end{array}\right]$
अतएव A (BC) = $\left[\begin{array}{rrr} 1 & 1 & -1 \\ 2 & 0 & 3 \\ 3 & -1 & 2 \end{array}\right]$$\left[\begin{array}{cccc} 7 & 2 & -3 & -1 \\ 4 & 0 & -4 & 2 \\ 7 & -2 & -11 & 8 \end{array}\right]$= ${\left[\begin{array}{cccc} 7+4-7 & 2+0+2 & -3-4+11 & -1+2-8 \\ 14+0+21 & 4+0-6 & -6+0-33 & -2+0+24 \\ 21-4+14 & 6+0-4 & -9+4-22 & -3-2+16 \end{array}\right]}$= ${\left[\begin{array}{llll} 4 & 4 & 4 & -7 \\ 35 & -2 & -39 & 22 \\ 31 & 2 & -27 & 11 \end{array}\right]}$
स्पष्टतया, (AB) C = A (BC)
(AB) (C) = $\left[\begin{array}{rc} 2 & 1 \\ -1 & 18 \\ 1 & 15 \end{array}\right]$ $\left[\begin{array}{rrrr} 1 & 2 & 3 & -4 \\ 2 & 0 & -2 & 1 \end{array}\right]$ = $\left[\begin{array}{crcc} 2+2 & 4+0 & 6-2 & -8+1 \\ -1+36 & -2+0 & -3-36 & 4+18 \\ 1+30 & 2+0 & 3-30 & -4+15 \end{array}\right]$
= $\left[\begin{array}{lrcc} 4 & 4 & 4 & -7 \\ 35 & -2 & -39 & 22 \\ 31 & 2 & -27 & 11 \end{array}\right]$
अब BC = $\left[\begin{array}{rr}1 & 3 \\ 0 & 2 \\ -1 & 4\end{array}\right]$ $\left[\begin{array}{rrrr} 1 & 2 & 3 & -4 \\ 2 & 0 & -2 & 1 \end{array}\right]$ = $\left[\begin{array}{rrrr} 1+6 & 2+0 & 3-6 & -4+3 \\ 0+4 & 0+0 & 0-4 & 0+2 \\ -1+8 & -2+0 & -3-8 & 4+4 \end{array}\right]$
= $\left[\begin{array}{cccc} 7 & 2 & -3 & -1 \\ 4 & 0 & -4 & 2 \\ 7 & -2 & -11 & 8 \end{array}\right]$
अतएव A (BC) = $\left[\begin{array}{rrr} 1 & 1 & -1 \\ 2 & 0 & 3 \\ 3 & -1 & 2 \end{array}\right]$$\left[\begin{array}{cccc} 7 & 2 & -3 & -1 \\ 4 & 0 & -4 & 2 \\ 7 & -2 & -11 & 8 \end{array}\right]$= ${\left[\begin{array}{cccc} 7+4-7 & 2+0+2 & -3-4+11 & -1+2-8 \\ 14+0+21 & 4+0-6 & -6+0-33 & -2+0+24 \\ 21-4+14 & 6+0-4 & -9+4-22 & -3-2+16 \end{array}\right]}$= ${\left[\begin{array}{llll} 4 & 4 & 4 & -7 \\ 35 & -2 & -39 & 22 \\ 31 & 2 & -27 & 11 \end{array}\right]}$
स्पष्टतया, (AB) C = A (BC)
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