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 nonsingular idempotent matrix. Then the sum of all the elements of the set S is .................

Answer 1

Given $A=\left[\begin{array}{ccc}{1}^2-3& p& 0\\ 0& m^2-8& q\\ r& 0& n^2-15\end{array}\right]$
Now,$A^2=\left[\begin{array}{ccc}{1}^2-3& p& 0\\ 0& m^2-8& q\\ r& 0& n^2-15\end{array}\right]\left[\begin{array}{ccc}{1}^2-3& p& 0\\ 0& m^2-8& q\\ r& 0& n^2-15\end{array}\right]$
$\Rightarrow A^2=\left[\begin{array}{ccc}(1^2-3{)}^2& p(l^2+m^2-11)& pq\\ qr& (m^2-8{)}^2& q(m^2+n^2-23)\\ r(l^2+m^2-8)& rp& (n^2-15{)}^2\end{array}\right]$
Also,given A is non-singular idempotent matrix.
$A^2=A$
$\Rightarrow \left[\begin{array}{ccc}(1^2-3{)}^2& p(l^2+m^2-11)& pq\\ qr& (m^2-8{)}^2& q(m^2+n^2-23)\\ r(l^2+m^2-8)& rp& (n^2-15{)}^2\end{array}\right]=\left[\begin{array}{ccc}{1}^2-3& p& 0\\ 0& m^2-8& q\\ r& 0& n^2-15\end{array}\right]$
$\Rightarrow (l^2-3)(l^2-4)=0\Rightarrow l=\pm 2,\pm \sqrt{3}$
$\Rightarrow (m^2-8)(m^2-9)=0\Rightarrow m=\pm 3,\pm 2\sqrt{2}$
$\Rightarrow (n^2-15)(n^2-16)=0\Rightarrow n=\pm 4,\pm \sqrt{15}$
Also, $pq=0\Rightarrow p=0orq=0$
$qr=0\Rightarrow q=0orr=0$
$pr=0\Rightarrow p=0orr=0$
Hence, $S=0,0,0\pm 2,\pm 3,\pm 4,\pm \sqrt{3},\pm 2\sqrt{2},\pm \sqrt{15}$
So, required sum =0