ATAR Notes: Forum

VCE Stuff => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Mathematics => Topic started by: QuantumJG on September 15, 2009, 08:21:18 pm

Title: Linear Algebra Questions!
Post by: QuantumJG on September 15, 2009, 08:21:18 pm: Idempotent matrices. A matrix P is called Idempotent if P^2 = P. If P is idempotent and P =/= I show that detP = 0.

I don't know where to start with this!!!

Determine the Eigenvalue of A if:

A = [2 2 1; 2 5 2; 1 2 2]

and then determine the Eigenvectors.

We haven't looked at Eigenvalues or Eigenvectors yet! This is just stuff for before we look a it.
Title: Re: Linear Algebra Questions!
Post by: kamil9876 on September 15, 2009, 08:38:29 pm: Idempotent question:

If $detP\neq 0$ then P is invertible.

Multiplying both sides by $P^{-1}$ .

$P^{-1} P^2=P^{-1} P$

$\implies P=I$

But this contradicts the hypothesis that $P\neq I$ Hence our assumption that $detP\neq 0$ is false and hence $detP=0$ .
Title: Re: Linear Algebra Questions!
Post by: QuantumJG on September 15, 2009, 08:40:53 pm: Quote from: kamil9876 on September 15, 2009, 08:38:29 pm
Idempotent question:

If $detP\neq 0$ then P is invertible.

Multiplying both sides by $P^{-1}$ .

$P^{-1} P^2=P^{-1} P$

$\implies P=I$

But this contradicts the hypothesis that $P\neq I$ Hence our assumption that $detP\neq 0$ is false and hence $detP=0$ .

Thanks

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