Solving with an inverse matrix but no b

Question

So here's what I'm looking at: I'm given two matrices, A and A^-1

$$ A = \pmatrix{I&0&0\\ A&I&0\\B&D&I}, \quad A^{-1} = \pmatrix{I & 0 & 0\\ P & I & 0\\ Q & R & I}. $$

I'm asked to solve for P, Q, and R. I'd expect to use x = A^-1b, but I am not given b.

I noticed that both matrices have the same upper-right form - I on the main diagonal and all zeros above. Is that significant?

Thank you.

Answer 1

Hint: Using block-matrix multiplication, compute the product $AA^{-1}$ in terms of $A,B,D,P,Q,R$. Using the fact that $AA^{-1}$ must be the identity matrix, we end up with a system of three equations that we can solve to write $P,Q,R$ in terms of $A,B,D$.

To answer your explicit question: yes, the structure of the matrix is significant since with a different structure, the resulting system might not have a solution. However, this fact does not need to be explicitly considered in solving the problem.