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 $X=\begin{bmatrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{bmatrix}$
$XX^{T}=\begin{bmatrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{bmatrix}\begin{bmatrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{1} & b_{2} & c_{3} \end{bmatrix}$
$=\begin{bmatrix} a_{1}^{2}+a_{2}^{2}+a_{3}^{2} & a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3} & a_{1}c_{1}+a_{2}c_{2}+a_{3}c_{3} \\ a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3} & b_{1}^{2}+b_{2}^{2}+b_{3}^{2} & b_{1}c_{1}+b_{2}c_{2}+b_{3}c_{3} \\ a_{1}c_{1}+a_{2}c_{2}+a_{3}c_{3} & b_{1}c_{1}+b_{2}c_{2}+b_{3}c_{3} & c_{1}^{2}+c_{2}^{2}+c_{3}^{2} \end{bmatrix}$
$Tr\left(X X^{T}\right)=a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+b_{1}^{2}+b_{2}^{2}+b_{3}^{2}+c_{1}^{2}+c_{2}^{2}+c_{3}^{2}=28$
There is only one possibility in which $7$ entries are $2$ and $2$ entries are $0$
Hence, the number of ways $=\_{}^{9}C_{7}=36$