Question

find the minimum polynomial for the following matrix
7 4 -4
A= 4 -8 1
-4 -1 -8

Answer 1

first you need the characteristic polynomial, if i remember correctly
that would be the determinant of \(xI-A\)
i would cheat
http://www.wolframalpha.com/input/?i=charcterisic+polynomial++ {{7%2C+4%2C+-4}%2C+{4%2C+-8%2C+1}%2C+{-4%2C-4%2C+-8}}

Answer 2

since it seems unlikely that this factors, i believe it is the minimal polynomial as well, although i could be wrong about that onr

Answer 3

i found the characteristic polynomial
and i also found the adjoint matrix ,
wat should i do after ?

Answer 4

ooops i put it in wrong

Answer 5

looks like does factor
\[-(\lambda +9)(\lambda^2-79)\]

Answer 6

i think for minimal poly, you check to see if one or the other of these factors will give the zero matrix

Answer 7

it is pretty clear that \(A+9\) is not the zero matrix, so i guess the only other possibility is that \(A^2-79\) i the zero matrix.
if not, then you have your minimum polynomial already.
bit rusty on this, but i think it is right

Answer 8

i guess i mean \(A+9I\) and \(A^2-79I\) but we can check the last one

Answer 9

how to determine the greatest common divisor of the adjoint ,
because one of the steps it said we must find the common divisor of the adj

Answer 10

that i do not know sorry