The total number of 3 $\times$ 3 matrices A having entries from the set {0, 1, 2, 3} such that the sum of all the diagonal entries of AAT is 9, is equal to _____________.
Answer (integer)
766
Solution
$$A{A^T} = \left[ {\matrix{
x & y & z \cr
a & b & c \cr
d & e & f \cr
} } \right]\left[ {\matrix{
x & a & d \cr
y & b & e \cr
z & c & f \cr
} } \right]$$<br><br>$$ = \left[ {\matrix{
{{x^2} + {y^2} + {z^2}} & {ax + by + cz} & {dx + ey + fz} \cr
{ax + by + cz} & {{a^2} + {b^2} + {c^2}} & {ad + be + cf} \cr
{dx + ey + fz} & {ad + be + cf} & {{d^2} + {e^2} + {f^2}} \cr
} } \right]$$<br><br>$$Tr(A{A^T}) = {x^2} + {y^2} + {z^2} + {a^2} + {b^2} + {c^2} + {d^2} + {e^2} + {f^2} = 9$$<br><br>Case-I : Nine ones = 1 case<br><br>Case-II : 8 zeroes and one entry is 3 = ${{{9!} \over {8!}} = 9}$ cases<br><br>Case-III : Two 2’s, one 1’s and 6 zeroes = ${{9!} \over {2!6!}} = 63 \times 4 = 252$<br><br>Case IV : one 2, five 1, rest 0 ${{9!} \over {5!3!}} = 63 \times 8 = 504$<br><br>$\therefore$ Total cases = 9 + 252 + 504 + 1 = 766
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
This question is part of PrepWiser's free JEE Main question bank. 274 more solved questions on Matrices and Determinants are available — start with the harder ones if your accuracy is >70%.