3×3m matrix then A^6 =?
Answers
Answer:
I am trying to find all matrices which solve the matrix equation
M2−3M+3I=0
M2−3M+3I=0
Since this doesn't factor I tried expanding this in terms of the coordinates of the matrix. It also occurs to me to put it into "vertex" form:
M2−3M+94I+34I=0
M2−3M+94I+34I=0
(M−32I)2=−34I
(M−32I)2=−34I
but this doesn't look much better.
What I found from expanding by coordinates was, if M=(acbd)M=(abcd) then
(a2+bc−3a+3ac+cd−3cab+bd−3bbc+d2−3d+3)=(0000)
(a2+bc−3a+3ab+bd−3bac+cd−3cbc+d2−3d+3)=(0000)
From the off-diagonal entries I get that either
a+d−3=0
a+d−3=0
or
b=c=0
b=c=0
If a+d−3≠0a+d−3≠0 then a2−3a+3=0a2−3a+3=0 and likewise for dd. Then we get more cases for aa and dd.
If a+d−3=0a+d−3=0 the upper-left is unchanged and the lower-right is
bc+(3−a)2−3(3−a)+3=0
bc+(3−a)2−3(3−a)+3=0
which simplifies to the same thing from the upper-left and so is redundant. In the off-diagonals
ac+c(a−3)−3c=0⇒
ac+c(a−3)−3c=0⇒
2ac−6c=0