Question

If A is a diagonal matrix of order 3 $\times$ 3 is commutative with every square matrix of order 3 $\times$ 3 under multiplication and tr(A) $=$ 12, then the value of $|A{|}^{1/2}$ is ..............

Answer 1

$letA=\left[\begin{array}{ccc}{d}_1& 0& 0\\ 0& d_2& 0\\ 0& 0& d_3\end{array}\right]andB=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]givenAB=BAand{d}_1+d_2+d_3=12\Rightarrow \left[\begin{array}{ccc}{d}_1& 0& 0\\ 0& d_2& 0\\ 0& 0& d_3\end{array}\right]\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]\left[\begin{array}{ccc}{d}_1& 0& 0\\ 0& d_2& 0\\ 0& 0& d_3\end{array}\right]comparingelementsonbothsidesgives{d}_1=d_2=d_3\left|A\right|=d_1{d}_2{d}_3=4^3=64Hence,{\left|A\right|}^{\frac{1}{2}}=8$