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

Correct Answer: 414

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 \(414\).