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 (integer)
540
Solution
$$\left[ {\matrix{
a & b & c \cr
d & e & f \cr
g & h & i \cr
} } \right]\left[ {\matrix{
a & d & g \cr
b & e & h \cr
c & f & i \cr
} } \right]$$<br><br>${a^2} + {b^2} + {c^2} + {d^2} + {e^2} + {f^2} + {g^2} + {h^2} + {i^2} = 7$<br><br><b>Case I :</b> Seven (1's) and two (0's)<br><br>Number of such matrices = ${}^9{C_2} = 36$<br><br><b>Case II :</b> One (2) and three (1's) and five (0's)<br><br>Number of such matrices = ${{9!} \over {5!3!}} = 504$<br><br>$\therefore$ Total = 540
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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