Problem 1 Matrix differentiation - . Let A(t) be an nxn matrir, which depends on t, i.c. its elements are functions of t: (A(0)lij = aij(t), for i, j = 1,...,n The derivative A(t) of A(t) with respect to t, is also an n xn matriz and is defined by taking element-uise derivatives: (Ä(t)ly = åij(t), for i, j = 1,...,n a) For n xn differentiable matrices A1(t), Ag(t) prove: (41(1) A2(t) = Å¡(t)Aa(t) + A1(t)Ž{t) Using induction, prove that for n x n differentiable matrices A1(t), A2(t), .. ALE), we have: The erponential of anxn matriz A is defined as the follouring: exp A = uhere A° = I. Suppose A(t) and Ä(t) commute. Prove: aexp A(t) = Â(t) exp A(t)

The concepts of linear and non-linear systems are a difficult topic. What concepts/terms (only the names) does one need to grasp in order to understand linear and non-linear systems? Give an example of the simplest linear system and non-linear system imaginable?

Now, let's discuss the difference cases of solution to a given system of linear equations. There are three different cases: a) The system has exactly one solution (one set of solution). b) The system has no solution c) The system has infinitely many solutions. In this assignment, you will need to give an example a system of two linear equations (2 equations, 2 unknowns) for the three cases.

I need help on this!