U: R­³ -> R³ , U(x, y, z) = (5x+8y+16z,4x+y+8z,-4x-4y-11z) is a lineer transformation. Find the functions that accept U as a root.

Answer 1

The linear transformation U in the matrix form is

⌈5 8 16⌉

|4 1 8|

⌊-4 -4 -11⌋.

The characteristic polynomial of a matrix always accept the matrix as its root, which is not hard to prove when you think about the jordan carnonical form of the matrix.

So all we need to do here is to find the characterstic polynomial of U. That is the determinant of the matrix

⌈x-5 -8 -16⌉

det|-4 x-1 -8| = (x-5)(x-1)(x+11) + 4∗4∗16 + 4∗8∗8 - (-16)(x-1)(4) - 4∗8∗(x+11) - (x-5)(4)(-8).

⌊4 4 x+11⌋.