Why Set of nxn Matrices w/ Zero Determinant Not Subspace of Mn,n

[yanjt]

Hi,
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!

 

[Billy Bob]

Not closed under matrix addition.

 

[JG89]

In general, the relation det(a + b) = det(a) + det(b) doesn't hold. Try to think of an example where det(a) = det(b) = 0 but det (a + b) is not equal to zero.

 

[matt grime]

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).

 

[yanjt]

Hi, I do not understand why it is not closed under matrix addition.It is still a M_n,n matrix isn't it?

 

[Billy Bob]

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

 

[matt grime]

Hi, I do not understand why it is not closed under matrix addition.It is still a M_n,n matrix isn't it?

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