How many `3xx3` matrices `M` with entries from `{0,1,2}` are there, for which the sum of the diagonal entries of `M^T Mi s5?` 126 (b) 198 (c) 162 (d) 135

Options

A. 198
B. 126
C. 135
D. 162

Answer

A

Explanation

Let `M=[(a,b,c),(d,e,f),(g,h,i)]`
`:. Tr(M^(T)M)=a^(2)+b^(2)+c^(2)+d^(2)+e^(2)+f^(2)+g^(2)+h^(2)+i^(2)`
`=5`, where entries are `{0, 1, 2}`
only two cases are possible
Case I : five entries are 1 and other four are 0.
`:.` Number of matrices `=.^(9)C_(5)xx1=126`
Case II : One entry is 2, one entry is 1 and others are 0.
`:.` Number of matrices `=.^(9)C_(2)xx2!=72`
So, total number of matrices `=126+72=198`