True or False: If $A^2$ is invertible, then $A$ is also invertible.
($A$ is a matrix here.)
The answer is true. I was trying to come up with an example that makes this false.
But I couldn't. Could anybody help me prove this?
[Math] If $A^2$ is invertible, then $A$ is also invertible
inverselinear algebramatrices
Best Answer
Hint: Suppose $B$ is the inverse of $A^2$. That is, let $B$ be the matrix such that $(A^2)\cdot B=I$ where $I$ is the identity matrix. Note that matrix multiplication is associative, so $$I=(A^2)\cdot B=(A\cdot A)\cdot B=A\cdot(A\cdot B).$$ Do you see the inverse to the matrix $A$?
I am implicitly using the fact that (for square matrices) a one-sided inverse, for either side, will also necessarily be a two-sided inverse. Here is the math.SE thread about this fact.