Question

For I =$\begin{bmatrix}1 & 0 \\\0 & 1 \end{bmatrix}$, if X and Y are square matrices of order 2 such that XY = X and Y X = Y , then (Y2 + 2Y ) equals to:

• A. 2Y
• B. I+3X
• C. I+3Y
• D. 3Y

Answer 1

Answer: (D)

3Y

Answer 2

The given conditions are:

$XY = X$ and $YX = Y$.
From $YX = Y$, we can factorize:
$Y(X-I) = 0$.
Thus, $Y=0$ or $X=I$. Since $X \neq 0$, we take $X=I$. Substituting into $Y^2+2Y$:
$Y^2 + 2Y = Y(Y+2)$.
From $YX = Y$, $YI = Y$, so $Y^2 = Y$. Substitute $Y^2 = Y$:

$Y^2 + 2Y = Y + 2Y = 3Y$.
Thus, $Y^2 + 2Y = 3Y$.