If A is a square matrix of order 3 and I is an ldentity matrix of order 3 such that `A^(3) - 2A^(2) - A + 2l =0,` then A is equal to

Options

A. I
B. 2I
C. `[[2 ,-1,2],[1,0,0],[0,1,0]]`
D. `[[2 ,1,-2],[1,0,0],[0,1,0]]`

Answer

A, B, D

Explanation

It is clear that A = I and ` A = 2I` satisfy the given
equaion `A^(3) - 2A^(2) - A + 2 I = 0`
and the characteristic
equation of the matrix in (c ) is
`[[2-lambda, -1 , 2 ],[1, -lambda, 0 ],[0 , 1, -lambda]]=0`
`rArr lambda^(3) - 2 lambda ^(2) + lambda - 2 = 0, `
giving `A^(3) - 2A ^(2) + - 2I = 0, `
`ne A^(3) - 2A^(3) - A + 2I = 0`
and the characteristic equation of the matris in (d) is
`[[2-lambda, -1 , 2 ],[1, -lambda, 0 ],[0 , 1, -lambda]]= 0`
`rArr lambda ^(3) - 2lambda ^(2) - lambda + 2 = 0, `
giving `A^(3) - 2 A^(2) - A+ 2I = 0`