JEE Advance - Mathematics (2017 - Paper 2 Offline - No. 3)
How many 3 $$ \times $$ 3 matrices M with entries from {0, 1, 2} are there, for which the sum of the diagonal entries of MTM is 5?
198
162
126
135
Explanation
Sum of diagonal entries of MTM is $$\sum\limits_{}^{} {a_i^2} $$
$$\sum\limits_{i = 1}^9 {a_i^2} = 5$$
Possibilities
I. 2, 1, 0, 0, 0, 0, 0, 0, 0, which gives $${{9!} \over {7!}}$$ matrices
II. 1, 1, 1, 1, 1, 0, 0, 0, 0, which gives $${{9!} \over {4!\, \times 5!}}$$ matrices
Total matrices = 9 $$ \times $$ 8 + 9 $$ \times $$ 7 $$ \times $$ 2 = 198
$$\sum\limits_{i = 1}^9 {a_i^2} = 5$$
Possibilities
I. 2, 1, 0, 0, 0, 0, 0, 0, 0, which gives $${{9!} \over {7!}}$$ matrices
II. 1, 1, 1, 1, 1, 0, 0, 0, 0, which gives $${{9!} \over {4!\, \times 5!}}$$ matrices
Total matrices = 9 $$ \times $$ 8 + 9 $$ \times $$ 7 $$ \times $$ 2 = 198
Comments (0)
