Question

Prove that If A is a square and there exist matrices B and C such that AB = I and CA = I, then B = C and A is invertible.

Answer 1

Question 1.

Prove that if $A$ is square and there exist matrices $B$ and $C$ such that $AB=I$ and $CA=I$, then $B=C$ and $A$ is invertible.

Solution. Since $A$ and $I$ are square and $AB=CA=I$, then $B$ and $C$ should be square of the same size. Multiplying $AB=I$ by $C$ on the left, using the associativity of product and the fact that $I$ is the identity matrix, we get

$CAB=C.$

Similarly $CA=I$, being multiplied by $B$ on the right, gives

$CAB=B.$

Thus, $B=C$. Therefore, $AB=BA=I$ and hence $B$ is the inverse of $A$. ∎