(I have started reading Michael Artin’s Algebra and I think uploading solutions of some problems, especially some of the starred ones, might be useful to others like me who are currently self-studying mathematics. If I have obtained the solution from some source, I will mention the source with the solution. Pointing out errors is always very welcome.)

Problem Statement: where each block is an matrix. Suppose that is invertible and that . Use block multiplication to prove that .

Solution:

We note that is invertible, and by expanding the determinant along its first row, we see that .

Then .

Take .

Then .

Clearly, . Expanding along its first column, we get .