1. The problem statement, all variables and given/known data Suppose A is an invertible mxm matrix, B is an invertible nxn matrix, and C is an arbitrary mxn matrix. Is the matrix M = A|C ---- O|B invertible? Solve with proof. Hint: Use block multiplication. Note: I'm not quite sure how to draw this matrix on the forums. It should me an (m+n)x(m+n) matrix with the entries of A in the top left, the entries of C in the top right, zeros in the bottom left, and the entries of B in the bottom right. I hope that's clear. Also, O is a matrix with all entries equal to zero. 2. Relevant equations For some mxm matrix M: M*M^-1 = M^-1*M = I_m 3. The attempt at a solution I showed in a different problem that a matrix composed like in this problem, but with C being all zeros is invertible, so my approach for this problem is to assume that the matrix has an inverse, and then show that all of C's entries must be 0. Let the matrix's inverse M^-1 be: A`|C` ------ O`|B` With A`, C`, O`, and B` being some matrices with the same dimensions as their non primed counterparts. Now multiply M^-1*M = A`A + C`O|A`C + C`B ---------------------- O`A + B`O|O`C + C`B Then multiply M*M^-1: AA` + CO`|AC` + CB` ---------------------- OA` + BO`|OC` + BB` This must be equal to the (m+n)x(m+n) identity matrix, which can be written as: I_m| O -------- O |I_n With this knowledge and the fact that any matrix times O = O, we can write the following equations: nxm matrices: O`A + B`O = OA` + BO` = O mxn matrices: A`C + C`B = AC` + CB` = O mxm matrices: CO` = O nxn matrices: O`C = O I also know: A*A` = A`*A = I_m B*B` = B`*B = I_n I've tried rearranging these a little bit, but nothing has come of it so far. Am I on the right track? If so, could somebody nudge me in the right direction? If not, how should I attack this problem? Thanks! Edit: I just realized that I can do the following O`A = O Multiply both sides by A` = A^-1 O`AA` = OA` O`I_m = O O` = O This doesn't really help me prove that C is all zeros though.