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 Post subject: nonsingular, idempotent matricesPosted: Fri, 4 Mar 2011 11:07:10 UTC
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I am a bit stuck on this question. Any help would be appreciated!

Calculate all non-singular, idempotent matrices of size 3 x 3.

Thanks.

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 Post subject: Posted: Fri, 4 Mar 2011 11:37:39 UTC
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Idempotent: A² = A

Because A is non-singular, its inverse exists. Therefore...

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 Post subject: Posted: Sat, 5 Mar 2011 03:00:25 UTC
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aswoods wrote:
Idempotent: A² = A

Because A is non-singular, its inverse exists. Therefore...
it is invertible? i'm not quite sure...

also, because its non-singular, its determinant does not equal to zero. but how do we calculate that when the matrix has no specific entries?

Last edited by asa.hoshi on Sat, 5 Mar 2011 03:04:20 UTC, edited 1 time in total.

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 Post subject: Posted: Sat, 5 Mar 2011 03:04:10 UTC
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asa.hoshi wrote:
aswoods wrote:
Idempotent: A² = A

Because A is non-singular, its inverse exists. Therefore...
it is invertible? i'm not quite sure...

It means you should apply to both sides.

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 Post subject: Posted: Sat, 5 Mar 2011 03:09:07 UTC
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asa.hoshi wrote:
aswoods wrote:
Idempotent: A² = A

Because A is non-singular, its inverse exists. Therefore...
it is invertible? i'm not quite sure...

It means you should apply to both sides.

how so?

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 Post subject: Posted: Sat, 5 Mar 2011 03:22:14 UTC
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asa.hoshi wrote:
asa.hoshi wrote:
aswoods wrote:
Idempotent: A² = A

Because A is non-singular, its inverse exists. Therefore...
it is invertible? i'm not quite sure...

It means you should apply to both sides.

how so?

Exactly what I say, multiply both sides by the inverse of A, it exists because it's nonsingular.

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 Post subject: Posted: Sat, 5 Mar 2011 03:30:08 UTC
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but it says matrices of size 3x3. how would i go about that though?

so i know its a square matrix.

Last edited by asa.hoshi on Sat, 5 Mar 2011 03:32:29 UTC, edited 1 time in total.

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 Post subject: Posted: Sat, 5 Mar 2011 03:31:32 UTC
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asa.hoshi wrote:
but it says matrices of size 3x3. how would i go about that though?

What do you mean? What does the size have to do with it? You can multiply matrices, can you not? So if then you can multiply both sides of this by the same thing to get the same result, right?

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 Post subject: Posted: Sat, 5 Mar 2011 03:38:03 UTC
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 Post subject: Posted: Sat, 5 Mar 2011 04:03:49 UTC
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asa.hoshi wrote:

Right, and there you go.

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 Post subject: Posted: Sat, 5 Mar 2011 07:18:38 UTC
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I'm still confused by the question...
"Calculate all non-singular, idempotent matrices..."

what does it mean by 'calculate all'

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 Post subject: Posted: Sat, 5 Mar 2011 09:12:40 UTC
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If A is idempotent, then A^2 = A.
Because A is nonsingular, its inverse exists. Thus:

In other words, the identity matrix is the only nonsingular idempotent matrix.

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 Post subject: Posted: Sat, 5 Mar 2011 21:00:29 UTC
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asa.hoshi wrote:
I'm still confused by the question...
"Calculate all non-singular, idempotent matrices..."

what does it mean by 'calculate all'

Exactly what it says, calculate, i.e. compute what the matrix must look like, i.e. what its entries must be, and the computation you did, and which Matt has duplicated is exactly that, you computed that it must be the identity.

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