Let PX = X( X)-1 and MX = In - PX.Then MX MX =
A)X(
X)-1 - PX.
C)In.
D)MX.

Question

Let PX = X(  X)-1  and MX = In - PX.Then MX MX =
A)X( 
X)-1 - PX. 
C)In. 
D)MX.

Quizplus · Accepted Answer

When we multiply a matrix by its inverse, we get the identity matrix. So, PX X PX-1 = In. Thus, MX = In - PX means that we are subtracting the identity matrix from PX, resulting in a matrix with diagonal entries equal to 0 and off-diagonal entries equal to the negative of the corresponding entry in PX. When we multiply MX by itself, the diagonal entries remain 0 and the off-diagonal entries become negative twice, giving us the same result as MX. Therefore, MX MX = MX2 = D)MX.