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**(1)**

Let

A =

2 0

4 1

B =

2 0 −4

3 −2 6

C =

5 0 0

0 −1 0

0 0 0

and let f(t) = t^2 - 5t + 2. Compute the following if possible.

(a) A^3

(b) C^2003

(e) f(A)

(g) We define the matrix exponential by the Taylor series:

e^C = I + C + 1/2! * C^2 + 1/3! * C^3 + · · · + 1/n! * Cn + · · · .

Calculate e^C

**(2)**

An n × n matrix S (with real entries) is called a square root of the n × n matrix A (with real entries), if S2 = A.

Find the square roots of the matrix

A=

1 3

0 1

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I don't have an idea on how to do the problems just posted, I can do the rest and those that I did not post, but I never learned #2 and I don't know how to take powers of matrix nor recall series. Would anyone be kind enough to explain how to do these problems. It would be very much appreciated. Thank you.