Proof: The inverse of the inverse matrix is the matrix.

Question

If $A$ is a square matrix such that it is not singular, then $(A^{-1})^{-1} = A$ How can I prove this property? I would appreciate it if somebody can help me.

Answer 1

Of course we know that for invertible $A$, we have that there exists an invertible $A^{-1}$ such that $AA^{-1} = A^{-1}A = I.$ We also use the fact that $(AB)^{-1} = B^{-1}A^{-1}$.

$$ (AA^{-1})^{-1} = I^{-1} = I$$

$$(A^{-1})^{-1}A^{-1}= I$$

$$[(A^{-1})^{-1}A^{-1}]A = IA$$

$$(A^{-1})^{-1}(A^{-1}A) = A$$

$$(A^{-1})^{-1} = A$$

Answer 2

By definition, $C$ is the inverse of the matrix $B=A^{-1}$ if and only if $BC = CB = I$.

Therefore, you can prove your property by showing that a product of a certain pair of matrices is equal to $I$.

Answer 3

If $A$ is an invertible matrix, then a matrix $B$ is its inverse iff $AB=I=BA$. Since $A^{-1}A=I=AA^{-1}$, the inverse of $A^{-1}$ is $A$.

Answer 4

By the definition of the inverse $$AA^{-1}=A^{-1}A=I.$$

This reads both ways: $A^{-1}$ is the inverse of $A$, or $A$ is the inverse of $A^{-1}$.

Answer 5

Suppose that there were two different inverse matrices: $B$ and $(A^{-1})^{-1}$.

Then we have $A^{-1}B = I = A^{-1}(A^{-1})^{-1}$. If we multiply by $B$ then we have:

$$BA^{-1}B = BI = BA^{-1}(A^{-1})^{-1}$$

However, $BA^{-1} = I$, so we have $B=B=(A^{-1})^{-1}$. Thus the inverse is unique.

Finally we need to find some inverse of $A^{-1}$. We know that $A A^{-1} = A^{-1} A = I$ by definition. Thus by the uniqueness above $A=(A^{-1})^{-1}$.

Answer 6

$XA^{-1}=A^{-1}X=I$ is known to have the solution $X=A$, hence $(A^{-1})^{-1}=A$.

Answer 7

There are really three possible issues here, so I'm going to try to deal with the question comprehensively. First, since most others are assuming this, I will start with the definition of an inverse matrix. Given a matrix $X$ ($n\times n$), a matrix $Y$ ($n\times n$) is an inverse for $X$ if and only if: $$XY=YX=I_n$$

This definition says "an inverse" and not "the inverse." That is an important distinction. We could prove one or more of the following statements:

1. The matrix $A$ is an inverse of the matrix $A^{-1}$.

This is proved directly from the definition. Assuming only that some matrix $A^{-1}$ is an inverse of $A$, we have by definition ($A$ plays the role of $X$, $A^{-1}$ plays the role of $Y$): $$AA^{-1}=A^{-1}A=I$$ and by the symmetric property of equality, we may write: $$A^{-1}A=AA^{-1}=I$$ which is the definition of $A$ being an inverse of $A^{-1}$ (where the roles of $A$ and $A^{-1}$ are now reversed, so that $A$ is in the place of $Y$ and $A^{-1}$ in that of $X$).

That is the proof. You seem to be concerned that you don't know how to begin, proving this from the definition of inverse, but that's literally all there is to it. You're just parsing the definition, applying a very simple property of equality, and then parsing the definition again to draw a slightly different conclusion.

It is foolhardy to assume that a more complicated proof is better here, especially one that relies on lemmas and properties that are themselves not axioms or definitions. That's just good practice, but in fact, it might be worse than that.

Although it is not necessarily the case here, depending on how you proved the statement $(AB)^{-1}=B^{-1}A^{-1}$, you could find yourself proving these lemmas in a circular fashion. For example, the proof from Wolfram MathWorld of $(AB)^{-1}=B^{-1}A^{-1}$ seems to rely on the fact that $A$ is an inverse for $A^{-1}$ when it left-multiplies both sides by $A$ in order to knock out an $A^{-1}$. (It is also assuming inverses are unique, which is not necessary, see part 2 below.)

Of course, it is trivial to note that this rule "works both ways" so it's just as easy to "cancel out" an $A$ by right- or left-multiplying by $A^{-1}$ as it is to do the opposite, but that's the whole point of what you're trying to prove here in the first place. Longer proofs requiring more "tools" are usually not better when there is a simple alternative.

2. For any two inverses $B$ and $C$ of a matrix $A$, $B=C$.

Because you are using the notation $A^{-1}$, I am under the impression you already know this, since that notation seems to presume the uniqueness of an inverse.

This is proved fairly easily. From the definition, we have: $$AB=BA=I$$ $$AC=CA=I$$ From this, we use the transitive property of equality to obtain something like: $$AC=AB$$ and right-multiplying both sides by $A$ and applying the associative law gives $C=B$.

Thus we may now refer to the inverse of $A$, which means that we can upgrade statement 1 to say "The matrix $A$ is the inverse of the matrix $A^{-1}$."

3. The right- and left-inverses of a matrix $A$ are unique and equal.

I won't prove this, since it's very clear you don't mention left- and right-inverses, but repeating part 2 for each side proves each is unique, and a bit more work proves they are in fact equal.