Given the matrix A = ((1,(−1),1),(0,2,( −1)),(2,3,0) ) and B= ((3,3,(−1)),((−2),(−2),1),((−4),(−5),2) ) find the matrix product AB and BA state the relationship between A and B find also the matrix product BM, where M= ((8),((−7)),(1) ) Hence solve the system of equations: x−y + z = 8, 2y −z =−7, 2x + 3y = 1.


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Question Number 75099 by Rio Michael last updated on 07/Dec/19

$G i v e n t h e m a t r i x$ $A = (\begin{matrix} 1 & - 1 & 1 \\ 0 & 2 & - 1 \\ 2 & 3 & 0 \end{matrix}) a n d B = (\begin{matrix} 3 & 3 & - 1 \\ - 2 & - 2 & 1 \\ - 4 & - 5 & 2 \end{matrix})$ $f i n d t h e m a t r i x p r o d u c t A B a n d B A$ $s t a t e t h e r e l a t i o n s h i p b e t w e e n A a n d B$ $f i n d a l s o t h e m a t r i x p r o d u c t B M, w h e r e M = (\begin{matrix} 8 \\ - 7 \\ 1 \end{matrix})$ $H e n c e s o l v e t h e s y s t e m o f e q u a t i o n s :$ $x - y + z = 8,$ $2 y - z = - 7,$ $2 x + 3 y = 1 .$
Commented by kaivan.ahmadi last updated on 07/Dec/19

Commented by kaivan.ahmadi last updated on 07/Dec/19

Commented by kaivan.ahmadi last updated on 07/Dec/19

Commented by Rio Michael last updated on 07/Dec/19

$n i c e s i r t h a n k s$
	Answered by Kunal12588 last updated on 07/Dec/19

	$A = [\begin{matrix} 1 & - 1 & 1 \\ 0 & 2 & - 1 \\ 2 & 3 & 0 \end{matrix}], B = [\begin{matrix} 3 & 3 & - 1 \\ - 2 & - 2 & 1 \\ - 4 & - 5 & 2 \end{matrix}]$ $A B = [\begin{matrix} 3 + 2 - 4 & 3 + 2 - 5 & - 1 - 1 + 2 \\ 0 - 4 + 4 & 0 - 4 + 5 & 0 + 2 - 2 \\ 6 - 6 + 0 & 6 - 6 + 0 & - 2 + 3 + 0 \end{matrix}]$ $\Rightarrow A B = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = I$ $B A = [\begin{matrix} 3 + 0 - 2 & - 3 + 6 - 3 & 3 - 3 + 0 \\ - 2 + 0 + 2 & 2 - 4 + 3 & - 2 + 2 + 0 \\ - 4 + 0 + 4 & 4 - 10 + 6 & - 4 + 5 + 0 \end{matrix}]$ $\Rightarrow B A = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = I$ $∵ A B = B A = I \Rightarrow A^{- 1} = B & B^{- 1} = A$ $M = [\begin{matrix} 8 \\ - 7 \\ 1 \end{matrix}]$ $B M = {[\begin{matrix} 3 & 3 & - 1 \\ - 2 & - 2 & 1 \\ - 4 & - 5 & 2 \end{matrix}]}_{3 \times 3} {[\begin{matrix} 8 \\ - 7 \\ 1 \end{matrix}]}_{3 \times 1}$ $\Rightarrow B M = [\begin{matrix} 24 - 21 - 1 \\ - 16 + 14 + 1 \\ - 32 + 35 + 2 \end{matrix}] = [\begin{matrix} 2 \\ - 1 \\ 5 \end{matrix}]$ $g i v e n {e q}^{n} c a n b e w r i t t e n a s$ $[\begin{matrix} 1 & - 1 & 1 \\ 0 & 2 & - 1 \\ 2 & 3 & 0 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 8 \\ - 7 \\ 1 \end{matrix}]$ $\Rightarrow A X = M; w i t h X = [\begin{matrix} x \\ y \\ z \end{matrix}]$ $\Rightarrow A^{- 1} A X = A^{- 1} M$ $\Rightarrow I X = A^{- 1} M$ $\Rightarrow X = B M [∵ A^{- 1} = B p r o v e d a b o v e]$ $\Rightarrow [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 2 \\ - 1 \\ 5 \end{matrix}]$ $\Rightarrow x = 2, y = - 1 & z = 5$
	Commented by Rio Michael last updated on 07/Dec/19

	$t h a n k s s o m u c h s i r$
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