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?
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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.
If matrix $A$ is a transformation which is not invertible, then applying $A$ twice to make $AA = A^2$ also cannot be invertible.
If $A$ is not invertible it means that $y = Ax$ applies a transformation to a space of vectors $x$ which irretrievably destroys information is needed to map the resulting vectors $y$ back to the original values $x$. $Ax$ is a "trap door": a "one way function".
There is nothing which can multiply $Ax$ to recover the lost information, let alone another $A$!
So it is impossible for $AA$ to be invertible without $A$ being invertible.
Think the other way:
If $A$ is not invertible, could $A^2$ be invertible ?
Invertible means bijective which is equivalent to injective or surjective.
If $A$ is not injective, could $A^2$ be injective ?
No, it means that there exist $x_1,x_2$ such that $Ax_1=Ax_2$. Just multiply it by $A$, then there exists $x_1,x_2$ such that $A^2 x_1=A^2x_2$ and $A^2$ is not injective. $A^2$ is not invertible.
In term of Kernel, $A$ injective means $Ker(A) \neq [o]$. As we clearly have $Ker(A) \subset Ker(BA)$ for all B, taking $B =A$, gives $Ker(A^2)\neq [o]$. $A^2$ is not injective. $A^2$ is not invertible.
If $A$ is not surjective, could $A^2$ be surjective ?
It means that there exist a vector y such that there is no x that solve $Ax=y$. So there is no x that solve $AAx=z$ with $z=Ay$. $A^2$ is not surjective. $A^2$ is not invertible.
We clearly have $Im(AB) \subset Im(A)$, so for $B=A$ and with that $A^2$ surjective means $Im(A^2) = \Re^n$. Then by inclusion $Im(A) = \Re^n$. Then A is surjective. $A^2$ is not invertible.
There is faster solutions but i thinks you always should keep in mind the equivalence between bijective, surjective and injective (in finite dimension).