Only by hand solution neededFor all partsTake your time Needed to be solved this question correctly in 60 minutes and get the thumbs up please solve correctly in the order to get positive feedback I need accuracy any how please Problem 3.Assume that A is a n x n real matrix. We define the exponential of A, as the series1e^= 1+A+A²where I is the n x n identity matrix.a) Show that forA+.where a and b are 2 real numbers, we haveeª(8=a= (8 %)1n!0 eb02).A" +...b) Show that if B € M₂ (R) is similar to a diagonal matrix D with a transition matrixdenoted by P (that is B = PDP ¹) theneB Pepp 1c) Show that if A and B are 2 matrices such that AB = BA, then eA¹B = eA eB

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Problem 3. Assume that A is a n x n real matrix. We define the exponential of A, as the series e^ = 1+A+ where I is the n x n identity matrix. a) Show that for 1 +²+ A = +. - (88) b where a and b are 2 real numbers, we have 1 n! A" +... e - (89). 0 eb b) Show that if B € M₂ (R) is similar to a diagonal matrix D with a transition matrix denoted by P (that is B = PDP 1) then eB Pepp 1. c) Show that if A and B are 2 matrices such that AB = BA, then eA¹B = eAB