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# Linear Algebra Quick True/False Watch

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1. (Original post by theed)
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(c) To prove that , I can say that:

A matrix is invertible if and only if it has non-zero determinant so .

Furthermore, is non-empty since since .

Let . Then . Now by the properties of determinants,

.

So and .

Now suppose and

Then .

Therefore .

Now since (set ) .

If and if

Then

so that .

If is a matrix of the same size of with arbitrary real entries

i.e. for for all

then and it follows that ( and hence having non-zero determinant guarantees it is invertible). Hence .

Now since and it follows that if then .

Therefore .

Let .

Then (say)

Note that the possibility that doesn't hurt since is odd for all and consequently the -th root is always real.

We have for , and .

Therefore and is the internal direct product of and

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Substituting

Setting in the first sum

Combining both sums:

.

Equating the series to 0 term-by-term gives the indicial equation

We also get the recurrence relation

In general,

which give 2 independent solutions for

..
2. (Original post by theed)

So the limits of the integral change from to

we have that .

Hence we have

...
3. Where have you said what A, B and G are?

Whatever G is, it looks like it might be Abelian so there's no need to check that subgroups are normal.

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Updated: February 12, 2011
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