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

Hint:

When dealing with matrix equations involving products like \( XY \) or \( YX \), factorization can be a helpful tool. In this case, we used the identity \( Y(X - I) = 0 \), which gave us two possible solutions. Once you find the values of the variables, be sure to substitute them into the original expressions to simplify further and verify the results. Matrix algebra can be tricky, but breaking down the problem step by step makes it more manageable.

Answer 1

The Correct Option is D

To determine \((Y^2 + 2Y)\) given the matrix equations \(XY = X\) and \(YX = Y\), we analyze the matrices \(X\) and \(Y\) in relation to the identity matrix \(I\). The given conditions are:

• \(XY = X\)
• \(YX = Y\)

To find \(Y^2\), we can left-multiply the equation \(YX = Y\) by \(Y\):

• \(Y(YX) = Y(Y)\)
• This results in \(Y^2X = Y^2\).
• Since \(YX = Y\), we can substitute to get \(Y = Y^2\). Therefore, \(Y^2 = Y\).

Now, we evaluate the expression \(Y^2 + 2Y\):

• Substituting \(Y^2 = Y\), the expression becomes:
• \((Y + 2Y) = 3Y\)

Consequently, the expression \(Y^2 + 2Y\) simplifies to: 3Y

The correct answer is therefore 3Y.