For $I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$, if X and Y are square matrices of order 2 such that $X Y=X$ and $Y X=Y$, then $\left(Y^2+2 Y\right)$ equals to:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → $3Y$
$XY=X,\;YX=Y.$
$Y^2=Y(XY)= (YX)Y = Y^2.$
$YX=Y \Rightarrow Y(X-I)=0.$
$XY=X \Rightarrow (Y-I)X=0.$
$YX=Y \Rightarrow Y^2=Y.$
$Y^2+2Y=Y+2Y=3Y.$