A block matrix is a matrix made up of smaller matrices arranged in blocks. This is not a formal definition, but the following example illustrates the concept: the matrix can be written as a block matrix with four blocks equal to the 2 x 2 identity matrix:
You can find more examples on page 74 of the course textbook. Show (in general or with a reasonably sized example) that you can multiply block matrices block by block, i.e., for A = [A11 A12; A21 A22] and B = [B11 B12; B21 B22], we have AB = [A11B11 + A12B21 A11B12 + A12B22; A21B11 + A22B21 A21B12 + A22B22], where the blocks Aij and Bij are the appropriate dimensions for matrix multiplication to be defined. If proving in general, you may assume the blocks are square and of the same dimension. If working with an example, make sure your matrix is at least 4 x 4 and that you explain how you're applying the definition of matrix multiplication. Use this to find an inverse for the matrix A.