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Question Number 216526 by Mingma last updated on 10/Feb/25

Answered by Wuji last updated on 10/Feb/25

let B= [((a b)),((c d)),((e f)) ] , C= [((3 4)),((−1 0)) ] : AB= [((1 3 2)),((0 −1 1)) ] [((a b)),((c d)),((e f)) ]= [((a+3c+2e b+3d+2f)),((−c −e −d +f )) ] AB=C ⇒ a+3c+2e=3 ⇒ b+3d+2f=4 ⇒ −c +e=−1 ⇒e=c−1 ⇒ −d +f=0 ⇒ f=d a+3c+2(c−1)=3 ⇒ a+5c=5 ⇒a=5−5c from b+3d+2f=4 b+3d+2d ⇒b+5d=4 ⇒b=4−5d {c=0 , d=0 arbitrary} then, a=5 , b=4 ,c=0,e=−1 , f=d=0 B= [(( 5 4)),(( 0 0)),((−1 0)) ] Checking if AB=C [((1 3 2)),((0 −1 1)) ] [((5 4)),((0 0)),((−1 0)) ]= [((1(5)+3(0)+2(−1) 1(4)+3(0)+2(0))),((0(5)+(−1)(0)+1(−1) 0(4)+(−1)(0)+1(0))) ] [((5 +0−2 4+0+0)),((0 +0−1 0+0+0)) ]= [((3 4)),((−1 0)) ]

letB=[abcdef],C=[34−10]:AB=[1320−11][abcdef]=[a+3c+2eb+3d+2f−c−e−d+f]AB=C⇒a+3c+2e=3⇒b+3d+2f=4⇒−c+e=−1⇒e=c−1⇒−d+f=0⇒f=da+3c+2(c−1)=3⇒a+5c=5⇒a=5−5cfromb+3d+2f=4b+3d+2d⇒b+5d=4⇒b=4−5d{c=0,d=0arbitrary}then,a=5,b=4,c=0,e=−1,f=d=0B=[5400−10]CheckingifAB=C[1320−11][5400−10]=[1(5)+3(0)+2(−1)1(4)+3(0)+2(0)0(5)+(−1)(0)+1(−1)0(4)+(−1)(0)+1(0)][5+0−24+0+00+0−10+0+0]=[34−10]

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