Let M be any 33 matrix with entries from the set 0,1,2.

Question

Let $M$ be any $3	imes3$ matrix with entries from the set $\{0,1,2\}$. The maximum number of such matrices, for which the sum of diagonal elements of $M^TM$ is seven, is

Answer: 540

Accepted Answer

$\operatorname{trace}(M^TM)=\sum$ of squares of all 9 entries $=7$.

Case 1: seven entries equal $1$ and two entries equal $0$ (squares: $1	imes7=7$): $\binom92=36$ arrangements.

Case 2: one entry equal $2$ (square $4$), three entries equal $1$ (square $1$ each, total $3$), and five entries equal $0$: $4+3=7$. Number of arrangements: $\dfrac{9!}{1!\,3!\,5!}=504$.

Total $=36+504=540$.

Solutions · 1