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 YX = Y, then \((Y^2 + 2Y)\) equals to:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 3Y
Since Y = YX, one Y in \(Y^2\) can be written as YX. By the associative law:
\(Y^2 = (YX)Y\)
\(= Y(XY)\)
\(= YX \quad (\text{since } XY = X)\)
\(= Y\)
Hence
\(Y^2 + 2Y = Y + 2Y = 3Y\)