Show that A=[5−13−2] satisfies the equation A2−3A−7I=0 and hence find A−1.

Question

Show that A=[5−1​3−2​] satisfies the equation A2−3A−7I=0 and hence find A−1.

Filo · Accepted Answer

We have, A=[5−1​3−2​] ∴A2=A.A=[5−1​3−2​][5−1​3−2​]=[25−3−5+2​15−6−3+4​]=[22−3​91​]3A=3[5−1​3−2​]=[15−3​9−6​]And 7I=7[00​01​]=[7 0​07​]∴A2−3A−7I=[22−3​91​]−[15−3​9−6​]− [7 0​07​]= [22−15−7 −3+3−0​9−9−01+6−7​]= [0 0​00​]=0 Hence proved.Since, A2−3A−7I=0⇒A−1[(A2)−3A−7I]=A−10⇒A−1AA−3A−1A−7A−1I=0[∵A−10=0]⇒IA−3I−7A−1=0[∵A−1A=I]⇒A−3I−7A−1=0  [∵A−1I=A−1]⇒−7A−1=−A+3I=[−5 1 ​−3 2​]+[3 0 ​0 3​]=[−2 1 ​−3 5​]∴A−1=7−1​[−2 1 ​−3 5​]

Question Text	Show that $A = [5 - 1 3 - 2]$ satisfies the equation $A^{2} - 3 A - 7 I = 0$ and hence find $A^{- 1}$ .
Answer Type	Text solution:1
Upvotes	150

Show that $A = [5 - 1 3 - 2]$ satisfies the equation $A^{2} - 3 A - 7 I = 0$ and hence find $A^{- 1}$ .

Text solutionVerified

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Show that A=[5−1​3−2​] satisfies the equation A2−3A−7I=0 and hence find A−1.

Text solutionVerified

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Show that $A = [5 - 1 3 - 2]$ satisfies the equation $A^{2} - 3 A - 7 I = 0$ and hence find $A^{- 1}$ .