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sdkans
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 Posted: Fri, 6 Nov 2009 05:56:17 UTC Post subject: Matrix Determinants |
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Suppose that M   can be written in the form
where A is a square matrix. Prove that det(M) = det (A).
I did it by determinant expansion on the second row, but apparently you're suppose to use induction to prove it... so now i'm not sure how to do it. |
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outermeasure
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| Posted: Fri, 6 Nov 2009 14:12:20 UTC Post subject: Re: Matrix Determinants |
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| sdkans wrote: | Suppose that M can be written in the form
where A is a square matrix. Prove that det(M) = det (A).
I did it by determinant expansion on the second row, but apparently you're suppose to use induction to prove it... so now i'm not sure how to do it. |
Which definition of determinant are you using?
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sdkans
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| Posted: Fri, 6 Nov 2009 14:40:34 UTC Post subject: |
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just the normal definition, i guess, i don't know any other definitions
. ie. a 2x2 matrix det is just one diagonal minus the other. |
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outermeasure
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| Posted: Fri, 6 Nov 2009 14:44:30 UTC Post subject: |
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| sdkans wrote: | just the normal definition, i guess, i don't know any other definitions
. ie. a 2x2 matrix det is just one diagonal minus the other. |
Which "normal" definition? There are at least two usual ways to define determinant (namely either the volume form, or the Leibniz formula).
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sdkans
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| Posted: Fri, 6 Nov 2009 15:37:41 UTC Post subject: |
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| i'm thinking both would work? we learned both in class, the instructor did not specify which we should use. basically, if we had a 2x2 matrix, row 1: a b row 2:c d. the det would be ac-bd. if it's a 3x3, you can either do it with cramers rule or by expansion of a row(cofactors) |
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outermeasure
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| Posted: Fri, 6 Nov 2009 15:54:02 UTC Post subject: |
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| sdkans wrote: | | i'm thinking both would work? we learned both in class, the instructor did not specify which we should use. basically, if we had a 2x2 matrix, row 1: a b row 2:c d. the det would be ac-bd. if it's a 3x3, you can either do it with cramers rule or by expansion of a row(cofactors) |
... and what happens to  for n>3?
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sdkans
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| Posted: Fri, 6 Nov 2009 15:57:51 UTC Post subject: |
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| you can find the determinant by some kind of permutation format. i guess that would be the Leibniz forumla. our instructor just mentioned it |
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Matt
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| Posted: Fri, 6 Nov 2009 17:33:52 UTC Post subject: Re: Matrix Determinants |
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| sdkans wrote: | | I did it by determinant expansion on the second row, but apparently you're suppose to use induction to prove it... so now i'm not sure how to do it. |
For the inductive step, you can expand along the last row. |
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