Determinant of matrix with paiwise orthogonal columns

Myth

14. (6) It is known that all columns of $A$ are pairwise orthogonal. Prove that $|\det A|$ equals to product of lengths of vector-columns.

Host topic http://www.artofproblemsolving.com/Forum/viewtopic.php?t=33762

**Posted:** Sat Apr 16, 2005 5:12 am

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Myth is out of here

liyi

$A^TA$ is a diagonal matrix, the $i$ -th diagonal element is $\|\alpha_i\|^2$ , where $\alpha_1,\dots,\alpha_n$ are the column vectors of $A$ .
Thus
$(\det A)^2 = \det (A^T A) = \prod \|\alpha_i\|^2$

**Posted:** Sat Apr 16, 2005 5:19 am

Myth

I think it was the easiest problem on the contest. Try other problems! I believe in you!

**Posted:** Sat Apr 16, 2005 5:25 am

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