A=⎣⎡2−11−12−1+1−12⎦⎤verify that A3−6A2+9A−4I=0. find A−1

Question

A=⎣⎡​2−11​−12−1​+1−12​⎦⎤​verify that A3−6A2+9A−4I=0. find A−1

Filo · Accepted Answer

A2=A×A=⎣⎡​2−11​−12−1​+1−12​⎦⎤​⎣⎡​2−11​−12−1​+1−12​⎦⎤​=⎣⎡​6−55​−56−5​5−56​⎦⎤​A3=A2×A=⎣⎡​6−55​−56−5​5−56​⎦⎤​⎣⎡​2−11​−12−1​+1−12​⎦⎤​=⎣⎡​22−2121​−2122−21​21−2122​⎦⎤​A3−6A2+9A−4I=⎣⎡​22−2121​−2122−21​21−2122​⎦⎤​−⎣⎡​36−3030​−3036−30​30−3036​⎦⎤​+⎣⎡​18−99​−918−19​9−918​⎦⎤​−⎣⎡​400​040​004​⎦⎤​=⎣⎡​000​000​000​⎦⎤​=0hence A3−6A2+9A−4I=0A3−6A2+9A=4Ipre multiply by A−1⇒A2−6A+9I=4A−1A−1=41​⎩⎨⎧​⎣⎡​6−55​−56−5​5−56​⎦⎤​−⎣⎡​12−66​−612−6​6−612​⎦⎤​+⎣⎡​900​090​009​⎦⎤​⎭⎬⎫​=41​⎣⎡​31−1​131​−113​⎦⎤​

Question Text	$A = ⎣ ⎡ 2 - 1 1 - 1 2 - 1 + 1 - 1 2 ⎦ ⎤$ verify that $A^{3} - 6 A^{2} + 9 A - 4 I = 0$ . find $A^{- 1}$
Answer Type	Text solution:1
Upvotes	150

$A = ⎣ ⎡ 2 - 1 1 - 1 2 - 1 + 1 - 1 2 ⎦ ⎤$
verify that $A^{3} - 6 A^{2} + 9 A - 4 I = 0$ . find $A^{- 1}$

Text solutionVerified

Practice questions from similar books

A=⎣⎡​2−11​−12−1​+1−12​⎦⎤​verify that A3−6A2+9A−4I=0. find A−1

Text solutionVerified

Practice questions from similar books

$A = ⎣ ⎡ 2 - 1 1 - 1 2 - 1 + 1 - 1 2 ⎦ ⎤$
verify that $A^{3} - 6 A^{2} + 9 A - 4 I = 0$ . find $A^{- 1}$