How would you go about simplifying this matrix multiplication into its simplest form?

Question

How would you simplify this matrix multiplication ?

(A(AB)^T (ABC)^-T ) ^T C

Answer 1

Here are a few theorems you might find useful when simplifying these types of expressions.

1. (AB)^T=B^TA^T
2. (AB)^-1=B^-1A^-1
3. AA^-1=A^-1A=I
4. (A^T)^-1=(A^-1)^T
5. (A^T)^T=A

Now to start simplifying...

(A(AB)^T (ABC)^-T ) ^T C

I will start with 1...

(AB^TA^T (ABC)^-T)^TC

Now using 2...

(AB^TA^T (C^-1B^-1A^-1)^T)^TC

Using 1 again...

(AB^TA^T A^-TB^-TC^-T)^TC

Using 3...

(AC^-T)^TC

Using 1 again...

(C^-T)^TA^TC

Using 5...

C^-1A^TC

We have finished simplifying.