Let S be the set containing all 33 matrices with entries from -1,0,1.

Question

Let $S$ be the set containing all $3	imes3$ matrices with entries from $\{-1,0,1\}$. The total number of matrices $A\in S$ such that the sum of all the diagonal elements of $A^TA$ is $6$ is \_\_\_\_\_\_\_\_.

Answer: 5376

Accepted Answer

$\operatorname{trace}(A^TA)$ equals the sum of squares of all 9 entries of $A$. With entries in $\{-1,0,1\}$, each square is $0$ or $1$, so this sum equals the number of non-zero entries. We need exactly $6$ non-zero entries (each $\pm1$) and $3$ zero entries.

Choose which $6$ of the $9$ positions are non-zero: $\binom96=84$ ways; each non-zero entry independently is $+1$ or $-1$: $2^6=64$ ways.

Total $=84	imes64=5376$.

Solutions · 1