Which of the following is not a property of invertible matrices if A and B are matrices of the same order?

The correct option is (AB)^-1 = A^-1 B^-1.

To understand why this is the correct answer, let us analyze the properties of invertible matrices $A$ and $B$ of the same order.

First, let us recall the definition of the inverse of a matrix product. The inverse of the product of two invertible matrices $A$ and $B$ is given by the reversal law:
$${\left(AB\right)}^{-1}=B^{-1}{A}^{-1}$$
This shows that the order of the matrices is reversed when taking the inverse of a product.

We can verify this reversal law by multiplying $AB$ by $B^{-1}{A}^{-1}$:
$$\left(AB\right)\left(B^{-1}{A}^{-1}\right)=A\left(B{B}^{-1}\right)A^{-1}$$
Since $B{B}^{-1}=I$ (where $I$ is the identity matrix), this simplifies to:
$$AI{A}^{-1}=A{A}^{-1}=I$$
Similarly, multiplying in the reverse order yields:
$$\left(B^{-1}{A}^{-1}\right)\left(AB\right)=B^{-1}\left(A^{-1}A\right)B=B^{-1}IB=B^{-1}B=I$$
This confirms that ${\left(AB\right)}^{-1}=B^{-1}{A}^{-1}$ is a valid property.

Consequently, the statement ${\left(AB\right)}^{-1}=A^{-1}{B}^{-1}$ is generally not true because matrix multiplication is not commutative in general (i.e., $AB\ne BA$).

Let us briefly review the other options to confirm their validity:
1. (AA^-1) = (A^-1 A) = I: This is the fundamental definition of a matrix inverse.
2. (AB)^-1 = B^-1 A^-1: As proven above, this is the correct reversal law for the inverse of a matrix product.
3. AB = BA = I: If two matrices $A$ and $B$ are inverses of each other, then their product in either order results in the identity matrix $I$.

Therefore, the only statement that is not a property of invertible matrices is (AB)^-1 = A^-1 B^-1.