# Determinant help

• Jan 15th 2010, 08:06 PM
Dranalion
Determinant help
Hello,

I need help with this theory-based problem:

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

\$\displaystyle
|AA^T| < 0\$

I should say, from the properties I know of, \$\displaystyle |A| = |A^T|\$, what can I gather from this information?
• Jan 15th 2010, 08:16 PM
Prove It
Quote:

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:

\$\displaystyle
det(AA^T) < 0\$

(use properties of determinants to guide you)

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

Any help is much appreciated,

Dranalion

Recall that \$\displaystyle |A|\$ is a real scalar.

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

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

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

\$\displaystyle |AA^T| = |A||A^T|\$

\$\displaystyle = |A||A|\$

\$\displaystyle = |A|^2\$.

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

\$\displaystyle |A|^2 \geq 0\$ for all \$\displaystyle A\$.
• Jan 15th 2010, 08:40 PM
Prove It
BTW I assume that the elements in \$\displaystyle A\$ are all Real.

Bringing in Complex elements may affect this result.
• Jan 16th 2010, 12:55 AM
HallsofIvy
Quote:

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

Bringing in Complex elements may affect this result.

Not if you replace \$\displaystyle A^T\$ by \$\displaystyle 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 \$\displaystyle A^T\$. For matrices over the complex numbers, the adjoint is \$\displaystyle A^*\$.