Finding the inverse of a $4×4$ matrix containing only letter values.

Question

The question: Find the inverse (if it exists) of the following $4 × 4$ matrix $M$:$$ \begin{bmatrix} a & -b & -c & -d\\ b & a & -d & c\\ c & d & a & -b\\ d & -c & b & a\\ \end{bmatrix}$$ In attempting to answer this question, I have receieved a tip from my professor to use the transformation of $M$ in order to find the inverse. Multiplying that, the matrix falls into another $4 × 4$ matrix with a form of the identity matrix but modified with polynomials (of course, since it's only letter values.) I'm wondering how to carry forward with this? I tried zero-ing out rows and columns in order to use cofactor expansion but I'm not finding myself getting anywhere. I appreciate any help.

Answer 1

can you complete the following equation? $$ MM^T= ? $$