If A=⎣⎡122212221⎦⎤ , then show that A2−4A−5I=0, where I and 0 are the unit matrix and the null matrix of order 3 , respectively. Use this result to find A−1.

Question

If A=⎣⎡​122​212​221​⎦⎤​ , then show that A2−4A−5I=0, where I and 0 are the unit matrix and the null matrix of order 3 , respectively. Use this result to find A−1.

Filo · Accepted Answer

Given A=⎣⎡​122​212​221​⎦⎤​ A2=⎣⎡​122​212​221​⎦⎤​⎣⎡​122​212​221​⎦⎤​ ∴A2=⎣⎡​1+4+42+2+42+4+2​2+2+44+1+44+2+2​2+4+24+2+24+4+1​⎦⎤​ ∴A2=⎣⎡​988​898​889​⎦⎤​ ∴A2−4A−5I=⎣⎡​988​898​889​⎦⎤​−4⎣⎡​122​212​221​⎦⎤​−5⎣⎡​100​010​001​⎦⎤​ =⎣⎡​988​898​889​⎦⎤​−⎣⎡​488​848​884​⎦⎤​−⎣⎡​500​050​005​⎦⎤​ =⎣⎡​9−4−58−8−08−8−0​8−8−09−4−58−8−0​8−8−08−8−09−4−5​⎦⎤​ =⎣⎡​000​000​000​⎦⎤​ ∴A2−4A−5I=0Multiply both sides by A−1, we get,A−4−5I.A−1=0∴A−4−5A−1=0∴5A−1=A−4I∴A−1=51​(A−4I) ∴A−1=51​⎩⎨⎧​⎣⎡​122​212​221​⎦⎤​−4⎣⎡​100​010​001​⎦⎤​ ⎭⎬⎫​ ∴A−1=51​⎩⎨⎧​⎣⎡​122​212​221​⎦⎤​−⎣⎡​400​040​004​⎦⎤​ ⎭⎬⎫​ ∴A−1=51​⎣⎡​−322​2−32​22−3​⎦⎤​

Question Text	If $A = ⎣ ⎡ 122212221 ⎦ ⎤$ , then show that $A^{2} - 4 A - 5 I = 0$ , where I and 0 are the unit matrix and the null matrix of order 3 , respectively. Use this result to find $A^{- 1}$ .
Answer Type	Text solution:1
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If $A = ⎣ ⎡ 122212221 ⎦ ⎤$ , then show that $A^{2} - 4 A - 5 I = 0$ , where I and 0 are the unit matrix and the null matrix of order 3 , respectively. Use this result to find $A^{- 1}$ .

Text solutionVerified

Practice questions from similar books

If A=⎣⎡​122​212​221​⎦⎤​ , then show that A2−4A−5I=0, where I and 0 are the unit matrix and the null matrix of order 3 , respectively. Use this result to find A−1.

Text solutionVerified

Practice questions from similar books

If $A = ⎣ ⎡ 122212221 ⎦ ⎤$ , then show that $A^{2} - 4 A - 5 I = 0$ , where I and 0 are the unit matrix and the null matrix of order 3 , respectively. Use this result to find $A^{- 1}$ .