Matrices concept confusion....

[Clara Chung]

In the photo that is a matrix B. B^3 + B = 2B^2 +2I from calculations. But why not B must be equal to 2I, the reason is as follow,
B(B^2 + I ) = 2(B^2 +I)
B(B^2 + I)(B^2 + I)^(-1) = 2I (B^2 + I)(B^2 + I)^(-1)
so B = 2I why is my concept wrong? please explain to me.

 

[phyzguy]

Your concept assumes that the matrix (B^2 + I)^-1 exists. But there is also the possibility that the matrix (B^2 + I) is singular and has no inverse, which is the case for your matrix B.