We have seen the relation between invertibility and the existence of unique solution, which has immediate implication on the size of the matrix. In addition to this, we have more properties for inverse transformations.
We start with a simple property.
This is the consequence of the symmetry of the definition of inverse: B is inverse of A ⇔ A is inverse of B. Here is another simple property.
It is important to note the order of multiplication.
A consequence of the two properties is that (Ak)-1 = (A-1)k for any positive k and invertible A. We denote (AK)-1 or (A-1)k by AK Combined with earlier notation (see the end of this lecture), we have defined AK for any invertible A and any integer k.
Proof (AB)(B-1A-1) = A(BB-1)A-1 = A(I)A-1 = AA-1 = I. Similarly, (B-1A-1)(AB) = I.
Proof Suppose B and C are inverses of A. Then B = IB = (CA)B = C(AB) = CI = C.
Finally, we assemble equivalent invertibility criterion for square matrices. (see here for the proof)
For a square matrix A, the following are equivalent
Note that in the original definition of inverse matrix, we need both the 2nd and the 3rd properties hold. Now we see that for a square matrix, they are equivalent. The following are the same criterion stated for linear transformations.
For a linear transformation T: Rn → Rn, the following are equivalent.
If we do not know apriori the dimensions on both sides of T, we may use the following criterion for invertibility.
For a linear transformation T: Rn → Rm, the following are equivalent.
Proof By considering the matrices associated to linear transformations, this result then implies that T is invertible ⇔ T is onto and one-to-one. Conversely, suppose T is onto and one-to-one, then this result shows m ≤ n and this result shows m ≥ n. Thus m = n, and we may use the previous criterion to conclude that T is invertible. This completes the proof of the equivalence between the first two statements.