JEE MAIN - Mathematics (2021 - 24th February Morning Shift - No. 21)
Let M be any 3 $$ \times $$ 3 matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which the sum of diagonal elements of MTM is seven, is ________.
Answer
540
Explanation
$$\left[ {\matrix{
a & b & c \cr
d & e & f \cr
g & h & i \cr
} } \right]\left[ {\matrix{
a & d & g \cr
b & e & h \cr
c & f & i \cr
} } \right]$$
$${a^2} + {b^2} + {c^2} + {d^2} + {e^2} + {f^2} + {g^2} + {h^2} + {i^2} = 7$$
Case I : Seven (1's) and two (0's)
Number of such matrices = $${}^9{C_2} = 36$$
Case II : One (2) and three (1's) and five (0's)
Number of such matrices = $${{9!} \over {5!3!}} = 504$$
$$ \therefore $$ Total = 540
$${a^2} + {b^2} + {c^2} + {d^2} + {e^2} + {f^2} + {g^2} + {h^2} + {i^2} = 7$$
Case I : Seven (1's) and two (0's)
Number of such matrices = $${}^9{C_2} = 36$$
Case II : One (2) and three (1's) and five (0's)
Number of such matrices = $${{9!} \over {5!3!}} = 504$$
$$ \therefore $$ Total = 540
Comments (0)
