Question

Let $A$ and $B$ are two matrices of same order $3\times 3$ given by $A=\left[\begin{array}{ccc}1& 3& \lambda +2\\ 2& 4& 6\\ 3& 5& 8\end{array}\right]$ and $B=\left[\begin{array}{ccc}3& 2& 4\\ 3& 2& 5\\ 2& 1& 4\end{array}\right].$
If $Tr(AB{)}^T+Tr(BA{)}^T=Tr\left(BA\right),$ then the value of $2\lambda$ equals to

• A. $103$
• B. $203$
• C. $-103$
• D. $-203$

Answer 1

The correct option is C $-103$

$A=\left[\begin{array}{ccc}1& 3& \lambda +2\\ 2& 4& 6\\ 3& 5& 8\end{array}\right],B=\left[\begin{array}{ccc}3& 2& 4\\ 3& 2& 5\\ 2& 1& 4\end{array}\right]AB=\left[\begin{array}{ccc}1& 3& \lambda +2\\ 2& 4& 6\\ 3& 5& 8\end{array}\right]\left[\begin{array}{ccc}3& 2& 4\\ 3& 2& 5\\ 2& 1& 4\end{array}\right]$ $=\left[\begin{array}{ccc}16+2\lambda & \lambda +10& 4\lambda +27\\ 30& 18& 52\\ 40& 24& 69\end{array}\right]\Rightarrow Tr\left(AB\right)=2\lambda +16+18+69=2\lambda +103$
Now, given $Tr(AB{)}^T+Tr(BA{)}^T=Tr\left(AB\right)$ $\Rightarrow Tr\left(AB\right)+Tr\left(BA\right)=Tr\left(AB\right)(\because Tr(A)=Tr\left(A^T\right),Tr(AB)=Tr(BA\left)\right)\Rightarrow Tr\left(AB\right)=0$ $\Rightarrow 2\lambda =-103$