matrices question

[fuzzy]

Find a 3 by 3 matrix A, whose 9 entries contain at least 5 different non zero integers, with the property that:A^3-7A = 6I

Hi everyone! This is a fairly straightforward problem but Im having problems remembering my first year subjects! If anyone can point me in the right direction id be sooooo grateful... im guessing i begin by doing the multiplication of A with A then with A again and then subtracting 7A and getting equations equal to either 6 or 0 depending. but im not even sure about that... feeling very rusty.

thanks so mch if you can help.

[maxal]

Originally Posted by fuzzy

Find a 3 by 3 matrix A, whose 9 entries contain at least 5 different non zero integers, with the property that:A^3-7A = 6I

According to Cayley-Hamilton Theorem any matrix with the characteristic polynomial x^3-7x-6 can serve as A. The roots of x^3-7x-6=0 are 3,-1,-2 and, thus, we can start with the matrix
[3 0 0]
[0 -1 0]
[0 0 -2]
Now make whatever additions/subtractions of the rows of this matrix (that will not affect its characteristic polynomial) to get more distinct non-zero elements. Say, if we add the doubled third row to the second, and then subtract the second row from the first, we will get
[3 1 4]
[0 -1 -4]
[0 0 -2]
which can be taken as A.