Problem 1: Suppose that A, B = Matnxn (R) and that A is known to be invertible such that (A-¹BA) 25 = 21 where I is the identity matrix. Compute the matrix B100 Problem 2: Consider the n x n matrix 100 ... 1 20 0 A = 1 2 3 0 0 1 2 3 ... n That is, this matrix has all zeros above the main diagonal. a) Find A¹. To do this first compute A-1 in the special case of n = 4 and show all steps in the row reductions. Then use this special case to conjecture an answer to A-1 in the general/nn case. Finally, demonstrate that your conjecture is the correct answer by performing the relevant matrix multiplication. b) For the same A as above and for 5- = solve the system A7-6. Hint: this can be done with a single multiplication.

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