by barnamah » Thu 23 June, 2011 15:09
to find the characteristic equation of the matrix `A=[[4,0,1],[-2,1,0],[-2,0,1]]` it is important to know that we are asked to to find the determinant of the matrix `|A-λ\I|`
this means to subtract `λ\I` should be subtracted from the matrix and then take the determinant of the result.
`λ\I` means `λ` multiplied by identity matrix `I=[[1,0,0],[0,1,0], [0,0,1]]` which will look as follow:
`λ\I=[[λ,0,0],[0,λ,0], [0,0,λ]]` as you can see if we subtract this matrix from A, it means to `A-λ\I=[[4,0,1],[-2,1,0],[-2,0,1]]-[[λ,0,0],[0,λ,0], [0,0,λ]]` which is
`A-λ\I=[[4-λ,0,1],[-2,1-λ,0],[-2,0,1-λ]]` but this is not our aim, we want to find the determinant so we want `|[4-λ,0,1],[-2,1-λ,0],[-2,0,1-λ]|`.
I'll use cofactor expansion method to find the determinant. Using the middle column which has 2 zeros is the best choice to do it with just one value. The important step is the to pick the right coefficient sign for the 1-λ and it would be `(-1)^(r+c)(1-λ)` where r=row and c=column of the 1-λ we picked. as we see that 1-λ is located in row 2 and column 2 so r=2 and c=2 so `(-1)^(2+2)(1-λ)=(-1)^4(1-λ)=+(1-λ)`
`|[4-λ,0,1],[-2,1-λ,0],[-2,0,1-λ]|=1-λ|[4-λ,1],[-2,1-λ]|=(1-λ) (( 4-λ)(1-λ)-(-2)(1))=(1-λ)[( 4-λ)(1-λ)+2]=`
`=(1-λ)[( 4-λ)(1-λ)+2]=(1-λ)[4-4λ-λ+λ^2+2]`
`=(1-λ)[6-5λ+λ^2]=6-5λ+λ^2-6λ+5λ^2-λ^3`
`=6-11λ+6λ^2-λ^3=-λ^3+6λ^2-11λ+6`
`:.` the characteristic equation is `-λ^3+6λ^2-11λ+6=0`
Explore and know. That is asked.