Q. 154.1( 27 Votes )

For the matrix

Show that A3– 6A2 + 5A + 11 I = O. Hence, find A–1.

Answer :

Here A2 = A.A =


And hence A3 = A. A2 =



A3– 6A2 + 5A + 11 I =




Thus, A3– 6A2 + 5A + 11 I = 0


Now, A3– 6A2 + 5A + 11 I = 0,


(A.A.A)- 6 (A.A) +5A = -11I


Post-multiply with A-1 on both sides-


(A.A.A.A-1)- 6 (A.A.A-1) +5A.A-1 = -11I. A-1


(A.A.I) – 6(A.I) + 5I = -11I. A-1 {since A.A-1 = I}


(A.A) – 6A +5I = -11A-1 {since X.I = X}





Hence


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