Let A be a 33 matrix having entries from the set -1,0,1.

Question

Let $A$ be a $3	imes3$ matrix having entries from the set $\{-1,0,1\}$. The number of all such matrices $A$ having sum of all entries equal to $5$ is

Answer: 414

Accepted Answer

Let $p$ entries be $1$, $q$ entries be $-1$, and $9-p-q$ entries be $0$, with $p-q=5$ and $0\le p,q$, $p+q\le9$.

Possible $(p,q)$: $(5,0),(6,1),(7,2),(8,3),(9,4)$ --- but $p+q\le9$ excludes $(9,4)$ (sum $13>9$), leaving $(5,0),(6,1),(7,2),(8,3)$.

Number of arrangements for each: $\dfrac{9!}{p!\,q!\,(9-p-q)!}$:
- $(5,0)$: $\dfrac{9!}{5!\,0!\,4!}=126$
- $(6,1)$: $\dfrac{9!}{6!\,1!\,2!}=252$
- $(7,2)$: $\dfrac{9!}{7!\,2!\,0!}=36$
- $(8,3)$: exceeds $9$ entries ($8+3=11>9$), not valid.

Total $=126+252+36=414$.

Solutions · 1