The matrix `A = [{:(1,2),(2,2):}]` satisfies which one of the following polynomial equations ?

Options

A. `A^(2)+3A + 2I = 0`
B. `A^(2)+3A - 2I = 0`
C. `A^(2)-3A - 2I = 0`
D. `A^(2)-3A + 2I = 0`

Answer

C

Explanation

Given that, `A=[{:(1,2),(2,2):}]`
`therefore" "A^(2)=[{:(1,2),(2,2):}][{:(1,2),(2,2):}]=[{:(1+4,2+4),(2+4,4+4):}]=[{:(5,6),(6,8):}]`
Let `A^(2) + xA + yI = 0` where x and y are constant.
`rArr [{:(5,6),(6,8):}]+[{:(x,2x),(2x,2x):}]+[{:(y,0),(0,y):}]=[{:(0,0),(0,0):}]`
`rArr [{:(5+x+y,6+2x),(6+2x,8+2x+y):}]=[{:(0,0),(0,0):}]`
So, `6 + 2x = 0" "rArr x =-3`
`5 + x + y = 0" "rArr y =-5 - x =-2`
`" "rArr A^(2)-3A-2I=0`