Question

Matrices P and Q satisfies $\mathrm{PQ}=Q^{-1}$, where $Q=\left(\begin{array}{cc}2& -1\\ 2& 0\end{array}\right]$. Find the value of K for which $\mathrm{KP}-2Q^{-1}+I=0.$

• A. 2
• B. 3
• C. 4
• D. 6

Answer 1

The correct option is A 2

$\mathrm{PQ}=Q^{-1}\mathrm{or}{\mathrm{PQ}}^2=I\left(\mathrm{post}-\mathrm{multiplying}\mathrm{both}\mathrm{side}\mathrm{by}Q\right)\mathrm{Now},\mathrm{KP}-2Q^{-1}+I=0\mathrm{or}\mathrm{KPQ}-2Q^{-1}Q+\mathrm{IQ}=0\left(\mathrm{post}-\mathrm{multiplying}\mathrm{both}\mathrm{side}\mathrm{by}Q\right)\Rightarrow \mathrm{KPQ}-2I+Q=0\Rightarrow {\mathrm{KPQ}}^2-2Q+Q^2=0\Rightarrow \mathrm{KI}-2Q+Q^2=0\mathrm{now},\mathrm{by}\mathrm{putting}\mathrm{the}\mathrm{value}\mathrm{of}Q\mathrm{and}I,\mathrm{we}\mathrm{get}-K\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]-2\left(\begin{array}{cc}2& -1\\ 2& 0\end{array}\right]+\left(\begin{array}{cc}2& -1\\ 2& 0\end{array}\right]\left(\begin{array}{cc}2& -1\\ 2& 0\end{array}\right]=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\Rightarrow \left(\begin{array}{cc}K& 0\\ 0& K\end{array}\right]-\left(\begin{array}{cc}4& -2\\ 4& 0\end{array}\right]+\left(\begin{array}{cc}2& -2\\ 4& -2\end{array}\right]=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\Rightarrow \left(\begin{array}{cc}K-2& 0\\ 0& K-2\end{array}\right]=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\Rightarrow K=2$