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 Post subject: Proof by Induction: Determinant of scalar times matrixPosted: Sun, 19 Sep 2010 07:11:49 UTC

Joined: Sun, 27 Jun 2010 20:50:11 UTC
Posts: 9
Location: New Mexico, USA
Hi Guys,

My instructor was unable to complete any inductive proofs for linear algebra on the board, he simply said if it's true for k, it clearly must be true for k+1 (writing down nothing) - leaving us scratching our heads.

I have no experience with inductive proofs.

In homework I'm now perplexed on solving , Show that

I know the basis for induction is for n=1,

And somehow I need to show that for n>1,

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 Post subject: Re: Proof by Induction: Determinant of scalar times matrixPosted: Sun, 19 Sep 2010 07:28:15 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
rossheed wrote:
Hi Guys,

My instructor was unable to complete any inductive proofs for linear algebra on the board, he simply said if it's true for k, it clearly must be true for k+1 (writing down nothing) - leaving us scratching our heads.

I have no experience with inductive proofs.

In homework I'm now perplexed on solving , Show that

I know the basis for induction is for n=1,

And somehow I need to show that for n>1,

Yuck, that is the wrong way to define determinant if you want to prove things.

Hint: what is ? So what is ? And it should be .

_________________

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 Post subject: Determinant of scalar times matrix proved, I thinkPosted: Sun, 19 Sep 2010 20:13:13 UTC

Joined: Sun, 27 Jun 2010 20:50:11 UTC
Posts: 9
Location: New Mexico, USA
Thanks outermeasure - I think I get the idea of proof by induction now. I'm changing the basis to a 2x2 since a 1x1 seems to be a trivial case, and gives little information on how the problem scales. That hint you gave me about writing it properly (because I had forgot to include alpha in the determinant equation) sparked my brain to figure out the rest. Many thanks and respect!

My proof:
Spoiler:
Basis:

For n=2,

For n=k, , and we assume, is true.

For n=k+1,

But since by the n=k statement above.

Factoring out ,

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