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

Question

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

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

Monjur Alahi

4 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - A::B::D
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`