Permutations and Combinations 1 Question 2
2. How many $3 \times 3$ matrices $M$ with entries from ${0,1,2}$ are there, for which the sum of the diagonal entries of $M^{T} M$ is 5 ?
(2017 Adv.)
(a) 198
(b) 162
(c) 126
(d) 135
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Answer:
Correct Answer: 2. (a)
Solution:
- Sum of diagonal entries of $M^{T} M$ is $\sum a _i^{2}$.
$$ \sum _{i=1}^{9} a _i^{2}=5 $$
Possibilities I. $2,1,0,0,0,0,0,0,0$, which gives $\frac{9 !}{7 !}$ matrices
II. $1,1,1,1,1,0,0,0$, 0 , which gives $\frac{9 !}{4 ! \times 5 !}$ matrices
Total matrices $=9 \times 8+9 \times 7 \times 2=198$
