Question #e0ca4

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Answer 1 · 2018-01-16T11:40:13

See below.

a)

If #A# is a #(3 xx 3)# matrix then it obeys the characteristic polynomial

#A^3+alpha A^2+beta A + gamma I_3 = 0# then multiplying by #A#

#A^4+alpha A^3+beta A^2+ gamma A = 0# and then if #A^4 = 0#

#alpha A^3+beta A^2+ gamma A = 0# multiplying again by #A#

#alpha A^4+beta A^3+gamma A^2=0# but #A^4 = 0# then

#beta A^3+gamma A^2=0# multiplied by #A#

#beta A^4 + gamma A^3 = 0# but #A^4 = 0# so we conclude:

If #A# is #(3 xx 3)# and #A^4 = 0# then if #gamma ne 0# follows also #A^3 = 0#

The item b) is left to the reader as an exercise.