Determinant help

**Dranalion** · January 15th 2010, 08:06 PM

Hello,

I need help with this theory-based problem:

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

$<br /> |AA^T| < 0$

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

**Prove It** · January 15th 2010, 08:16 PM

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:

$<br /> det(AA^T) < 0$

(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 $A$ .

**Prove It** · January 15th 2010, 08:40 PM

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

Bringing in Complex elements may affect this result.

**HallsofIvy** · January 16th 2010, 12:55 AM

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 $A^*$ .

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