1. In this problem we will differentiate the inverse of an n x in matrix. We can think of n X 1?. matrices as points in R712. Dene f(A) = A'1 on the domain of all invertible n x 7: matrices. If B is n x n and close to the allU matrix we have a linear apprmdmation of f(A + B) of the form g{A + B) = A'l + DfA[B), where DfA : Eng > R"2 is a linear function. Rather than give formulae for all 114 matrix entries, we nd a way to write D fA in terms of matrix operations. (a) Now choose an n x 71 matrix B. Explain why %[(A+tB)f(A + tB]] is the allzero n x 11. matrix when t = 0. (13) Now explain why a + mama + ten 2 gm + mum + am] at a = 0. (Hint: Use the product rule for matrixvalued functions 13(3), ill/{(33): D(LM)$(W) = DLE(W)M(1') + L(I)DM$(W).) (c) Now nd the unique matrix U such that KA + tBMA'I + tU)] = U, in terms of A and B. {This is the only computation! Take care to remember matrix multiplication is not commutative.) Write down a formula for U as a function of A and B and check that (Hg is indeed linear. (Note: When n = 1 you should recover something familiar from single variable calculus