Since A and B are invertible matrices of order 3, it means that they have the same number of rows and columns, which is 3. This also implies that they are square matrices.

Let's denote the inverse of matrix A as A^(-1) and the inverse of matrix B as B^(-1).

1. Matrix A^(-1) exists:
Since A is an invertible matrix, it means that its determinant is non-zero (det(A) ≠ 0). This is a necessary condition for a matrix to have an inverse. Therefore, A^(-1) exists.

2. Matrix B^(-1) exists:
Similarly, B is an invertible matrix, so its determinant is non-zero (det(B) ≠ 0). Hence, B^(-1) exists.

Now, we can perform various operations with these two matrices:

1. Matrix Addition:
A + B is also a 3x3 matrix. The sum of two invertible matrices is again an invertible matrix.

2. Scalar Multiplication:
kA and kB (where k is a scalar) are also 3x3 matrices. The scalar multiple of an invertible matrix is again an invertible matrix.

3. Matrix Multiplication:
AB and BA are both 3x3 matrices. The product of two invertible matrices is again an invertible matrix.

4. Inverse of Matrix Addition:
(A + B)^(-1) exists. In general, the inverse of a sum of matrices is not equal to the sum of the inverses of the matrices.

5. Inverse of Scalar Multiplication:
(kA)^(-1) exists for any non-zero scalar k. The inverse of a scalar multiple of a matrix is equal to the reciprocal of the scalar multiplied by the inverse of the matrix.

6. Inverse of Matrix Multiplication:
(AB)^(-1) exists, and it is equal to B^(-1)A^(-1). In general, the inverse of a product of matrices is not equal to the product of the inverses of the matrices.

These are some of the operations and properties that can be applied to invertible matrices A and B of order 3.