The number of all 33 matrices A, with entries from the set -1,0,1, such that the sum of th…

Question

The number of all $3	imes3$ matrices $A$, with entries from the set $\{-1,0,1\}$, such that the sum of the diagonal elements of $AA^T$ is 3, is\_\_\_\_\_\_.

Answer: 672

Accepted Answer

$\operatorname{trace}(AA^T)$ equals the sum of squares of the 9 entries, so we need exactly $3$ entries equal to $\pm1$ and $6$ entries equal to $0$.

Choose which $3$ of the $9$ positions are non-zero: $\binom93=84$ ways; each independently $+1$ or $-1$: $2^3=8$ ways.

Total $=84	imes8=672$.

Solutions · 1