Question

The number of matrices $X$ with entries $\left\{0,2,3\right\}$ for which the sum of all the principal diagonal elements of $X\cdot X^T$ is $28$ (where $X^T$ is the transpose matrix of $X$ ), is

• A. $12$
• B. $18$
• C. $36$
• D. $44$

Answer 1

Answer: (C)

Let $X=\left[\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right]$
$X{X}^T=\left[\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right]\left[\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_1& b_2& c_3\end{array}\right]$
$=\left[\begin{array}{ccc}{a}_1^2+a_2^2+a_3^2& a_1{b}_1+a_2{b}_2+a_3{b}_3& a_1{c}_1+a_2{c}_2+a_3{c}_3\\ a_1{b}_1+a_2{b}_2+a_3{b}_3& b_1^2+b_2^2+b_3^2& b_1{c}_1+b_2{c}_2+b_3{c}_3\\ a_1{c}_1+a_2{c}_2+a_3{c}_3& b_1{c}_1+b_2{c}_2+b_3{c}_3& c_1^2+c_2^2+c_3^2\end{array}\right]$
$Tr\left(X{X}^T\right)=a_1^2+a_2^2+a_3^2+b_1^2+b_2^2+b_3^2+c_1^2+c_2^2+c_3^2=28$
There is only one possibility in which $7$ entries are $2$ and $2$ entries are $0$
Hence, the number of ways $=\_{}^9{C}_7=36$