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CBSE Class 12 · 2005 · MatricesVery Short Answer

If

A = [2132]

, prove that

A^{3} - 4 A^{2} + A = 0

.

Answer: Proved

Solution

The characteristic equation of

A

comes from its trace and determinant:

tr (A) = 4

and

det (A) = 4 - 3 = 1

. By the Cayley-Hamilton theorem

A

satisfies its own characteristic equation:

A^{2} - 4 A + I = 0 \Rightarrow A^{2} = 4 A - I .

Multiply by

A

to get

A^{3} = 4 A^{2} - A

. Substitute into the required expression:

A^{3} - 4 A^{2} + A = (4 A^{2} - A) - 4 A^{2} + A = 0.

Hence

A^{3} - 4 A^{2} + A = 0

. Proved.

CBSE Class 12 Board · 2005 · Subjective

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