Inverses of matrices and its properties.

Question

I'm reading in my textbook this theorem about the properties of inverse matrices:

I don't follow how 1) and 3) follow from the idea that inverses are unique. Why does the fact that a matrix only has one inverse matter in these theorems?

This is the theorem about uniqueness that I read:

Quick question, why does $CI = C$ in the proof for uniqueness?

Answer 1

Identity matrix is neutral with respect to multiplication, i.e. for any square matrix $P$ you have $PI=IP=P$. You can easily show this property, just put the definition of $I$ in the the definition of matrix multiplication.

For (1), suppose that $AB=BA=I$. By definition, it means that $A = B^{-1}$. Then again, if you put $B = A^{-1}$, then our hypothesis holds ($AA^{-1}=A^{-1}A=I$), hence $A = B^{-1} = (A^{-1})^{-1}$.

The (3) is done likewise.

Answer 2

The answer to the quick question is: because that's a property of the identity matrix. Just do the computation.

Concerning 1), you have $A.A^{-1}=\operatorname{Id}$. Since the inverse of $A^{-1}$ is the only matrix whose product by $A^{-1}$ is $\operatorname{Id}$ and since $A$ has the property, $(A^{-1})^{-1}=A$.

The case of 3) is similar. You have $\left(\frac1cA^{-1}\right).cA=\frac1ccA^{-1}A=\operatorname{Id}$. Since the inverse of $cA$ is the only matrix whose product by $cA$ is $\operatorname{Id}$ and since $\frac1cA^{-1}$ has the property, $(cA)^{-1}=\frac1cA^{-a}$.