Question

If X and Y are two singular matrices such that XY = Y and Y X = X, then X^2 + Y^2 equals (A) X + Y (B) XY (C) Y X (D) 2(X + Y )

Answer 1

XY=Y and YX=X
X²+Y²
=XX+YY
=(YX)(YX)+(XY)(XY)
=Y(XY)X+X(YX)Y
=Y(YX)+X(XY)
=YX+XY
=X+Y
Option (A) is the right answer.

Answer 2

Answer:

$X ^ { 2 } + Y ^ { 2 } = X + Y$

Hence the correct option is (A).

To find:

The value of $X ^ { 2 } + Y ^ { 2 }$

where X and Y are two singular matrices

Solution:

$X ^ { 2 } + Y ^ { 2 } = X \times X + Y \times Y$

Replace X with YX and Y with XY

$\begin{array} { c } { X ^ { 2 } + Y ^ { 2 } = Y X \times X + X Y \times Y } \\\\ { X ^ { 2 } + Y ^ { 2 } = Y X + X Y } \\\\ { X ^ { 2 } + Y ^ { 2 } = X + Y } \end{array}$

Hence, $X ^ { 2 } + Y ^ { 2 } = X + Y$

Hence the correct option is (A).