Question

I'm specifically stuck on part d. I have been attempting to prove it using the ratio test.

6. Recall that we may define the exponential function in terms of it's Taylor series expansion Similarly, we define the matrix exponential in terms of the following series 44 k=0 Where Ak is matrix multiplication of A with itself k times. Compute the matrix exponential for the following matricies (a) (1 point) 12. (b) (1 point) (1 =1) (c) (1 point) Any 2 x 2 nilpotent matrix (recall homework 2) (d) (1 point) Will every 2 x 2 matrix have a convergent matrix exponential? Why or why not? (Recall your Taylor series from 21C).

Answer 1

Aus wel Curen that :) Dow, Pok = I2 &k=0,1,2. = I2 - & Po 는 sow.po Eta De Ao eE c-o ان * bi 2) A ? (-:10:- = A² = AXA AKO Jl>2. A KLAK = Ao ca ko 13 PQ+A - Para I! (A -Pa) » eA=I2+A A el since any nilpotent makin û allowed.

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