1. ## Determinant help

Hello,

I need help with this theory-based problem:

Why is it not possible to find a square matrix such that:

$
|AA^T| < 0$

I should say, from the properties I know of, $|A| = |A^T|$, what can I gather from this information?

2. Originally Posted by Dranalion
Hello,

I need help with this theory-based problem:

Why is it not possible to find a square matrix such that:

$
det(AA^T) < 0$

(use properties of determinants to guide you)

I should say, from the properties I know of, $det(A) = det(A^T)$, what can I gather from this information?

Any help is much appreciated,

Dranalion
Recall that $|A|$ is a real scalar.

You should know that $|AB| = |A||B|$.

You should also know that $|A^T| = |A|$.

So if you let $B = A^T$ you have

$|AA^T| = |A||A^T|$

$= |A||A|$

$= |A|^2$.

Since $|A|$ is a real number, and the square of any real number is nonnegative, this means that

$|A|^2 \geq 0$ for all $A$.

3. BTW I assume that the elements in $A$ are all Real.

Bringing in Complex elements may affect this result.

4. Originally Posted by Prove It
BTW I assume that the elements in $A$ are all Real.

Bringing in Complex elements may affect this result.
Not if you replace $A^T$ by $A^*$- that is, in addition to interchanging rows and columns, you also take the complex conjugate of each number. The "adjoint" of A for matrices over the real numbers is $A^T$. For matrices over the complex numbers, the adjoint is $A^*$.