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# Tag Archives: Linear Algebra

## Zero determinant

Consider the real numbers for . Prove that

Solution

Using the identity in combination with we have:

## Zero determinant

Let such that

Prove that if is odd.

Solution

Let . Then

Taking conjugate transpose we also have that

Hence . However it also holds . Combiming these two we get that

If we are done. Otherwise is real. In that case we have

since is odd. Hence as wanted.

## Determinant of linearly independent vectors

Let be unitary linearly independent vectors. Evaluate the determinant

where denotes the outer product.

## Determinant

Let and  such that . Prove that

Solution

Note that

Since is real, its complex eigenvalues come in conjugate pairs. Thus, in this case we conclude that has eigenvalues .

Now, if is an eigenvalue of , then is an eigenvalue of . Thus, the matrix has eigenvalues and .

Now, is the product of these eigenvalues, which is to say

as desired.

## Power of matrix

Let . Prove that .

Solution

The characteristic polynomial of is . This in return means and . Thus,