Question

A real n x n matrix $A=\left[a_{ij}\right]$ is defined as follows $\{\begin{array}{cc}{a}_{ij}=i& \forall i=j\\ =0,& otherwise\end{array}$
The sum of all n eigen values of A is

• A. $\frac{n(n+1)}{2}$
• B. $\frac{n(n-1)}{2}$
• C. $\frac{n(n+1)(2n+1)}{2}$
• D. $n^2$

Answer 1

The correct option is A $\frac{n(n+1)}{2}$

${\left[\begin{array}{ccccccc}1& 0& 0& \cdots & \cdots & \cdots & 0\\ 0& 2& 0& \cdots & \cdots & \cdots & 0\\ 0& 0& 3& \cdots & \cdots & \cdots & 0\\ ⋮& & & & & & \\ ⋮& & & & & & \\ 0& 0& 0& \cdots & \cdots & \cdots & n\end{array}\right]}_{n\times n}$
= diagonal Matix

$\Rightarrow$ Eigen values of A are the corresponding diagonal elements i.e. 1,2,3,......, n
$\therefore$ Sum of all eigen values $1+2+3+....+n$
$=\frac{n(n+1)}{2}$