Question

Let $S$ be the set which contains all possible values of $l,m,n,p,q,r$ for which
$A=\left[\begin{array}{ccc}{l}^2-3& p& 0\\ 0& m^2-8& q\\ r& 0& n^2-15\end{array}\right]$ be a non singular idempotent matrix. Then the sum of all the elements of the set $S$ is

Answer 1

Given $A=\left[\begin{array}{ccc}{l}^2-3& p& 0\\ 0& m^2-8& q\\ r& 0& n^2-15\end{array}\right]$
$A$ is idempotent
$\Rightarrow A^2=A$
Also, given that $A$ is non-singular idempotent matrix. Thus $A^{-1}$ exists
$\Rightarrow A^{-1}$ pre multiplication is possible
$\Rightarrow A^{-1}\cdot A\cdot A=A^{-1}\cdot A\Rightarrow I\cdot A=I\Rightarrow A=I$
$\Rightarrow$ Non-singular Idempotent matrix will always be a unit matrix
Thus, By comapring $A$ with $I$
$l^2-3=1,p=0m^2-8=1,q=0n^2-15=1,r=0$
$\Rightarrow l=\pm 2,m=\pm 3,n=\pm 4$
$\Rightarrow l+m+n+p+q+r=0$