- #1

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I would like to know why the set of all n*n matrix whose determinant is zero is not a subspace of M

_{n,n}.Can anyone explain the reason for me?

Thanks!

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- Thread starter yanjt
- Start date

- #1

- 14

- 0

I would like to know why the set of all n*n matrix whose determinant is zero is not a subspace of M

Thanks!

- #2

- 392

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Not closed under matrix addition.

- #3

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- #4

matt grime

Science Advisor

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In general, the relation det(a + b) = det(a) + det(b) doesn't hold.

True, but not the point. If det were a linear map then its kernel is a subspace. Its kernel not being a subspace implies det is not a linear map. But that does not mean that its kernel is not a subspace (exercise: find an example of a map f from a vector space to the base field so that f is not linear, but the set of x such that f(x) is zero is a subspace).

- #5

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- #6

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Find two matrices whose det is 0 but whose sum has nonzero det.

- #7

matt grime

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_{n,n}matrix isn't it?

But that isn't the only criterion you had to have the sum satisfy, is it?

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