Math Problem, Mathematics Jordan Normal form

Project description
The problems see attachment. Don’t need to be 100% percent correct, but please give detailed proofs for the prove problems. Since it’s hard to type the signs, it can be a picture of hand-written, thanks.

Problem 1 ( 8 points) .
Determine the real Jordan normal form of the matrix

Problem 2 (2 +4 +3 +3 points).
Let A ? Mat ( 3: R ) be a matrix with

a) Show that det (A )? { ± 1}.
b) Prove that, for every eigenvalue ? ? C,

c ) Prove that the matrix A has an eigenvalue ? ? { ± 1} and
Det (A) = 1 = ? 1 applies. What about the reverse ?
d) Specify the real Jordan normal form of A .
Notice . The complex numbers on the amount one are exactly the complex numbers of the form
exp ( i • t) = cos ( t) + i • sin (t) t ? R.

Problem 3 (7 +5 points).
Let K be a field and N ? Mat ( n , K ) is a nilpotent matrix. a) Show that En – N is invertible and that:

b ) Calculate the the inverse of the matrix

Problem 4 (8 points) .
Give an example for (2 × 2) – matrices A, B ? Mat ( 2, C) with

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