Question

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

Answer 1

Let matrix $\begin{bmatrix} a & b & c \\\[0.3em] d & e & f \\\[0.3em] g & h & i \end{bmatrix}$
Given, $a + b + c + d + e + f + g + h + i = 5$

Possible cases	Number of ways
$5 → 1’$s, $4 →$ zeroes	$\frac {9!}{5!4!}=126$
$6 → 1’$s, $2 →$ zeroes, $1 →–1$	$\frac {9!}{6!2!}=252$
$7 → 1’$s, $2 →–1'$s	$\frac {9!}{7!2!}=36$

Total number of ways $= 126 + 252 + 36 = 414$

So, the answer is $414$.