Q. 154.1( 27 Votes )

# For the matrix

Show that A^{3}– 6A^{2} + 5A + 11 I = O. Hence, find A^{–1}.

Answer :

Here A^{2} = A.A =

And hence A^{3} = A. A^{2} =

∴ A^{3}– 6A^{2} + 5A + 11 I =

Thus, A^{3}– 6A^{2} + 5A + 11 I = 0

Now, A^{3}– 6A^{2} + 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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