1. is Derogatory Matrix ?
`[[3,10,5],[-2,-3,-4],[3,5,7]]`
Solution:
| `A` | = | | `3` | `10` | `5` | | | `-2` | `-3` | `-4` | | | `3` | `5` | `7` | |
|
A matrix is derogatory if it's eigenvalues are repeated.
`|A-lamdaI|=0`
| `(3-lamda)` | `10` | `5` | | | `-2` | `(-3-lamda)` | `-4` | | | `3` | `5` | `(7-lamda)` | |
| = 0 |
`:.(3-lamda)((-3-lamda) × (7-lamda) - (-4) × 5)-10((-2) × (7-lamda) - (-4) × 3)+5((-2) × 5 - (-3-lamda) × 3)=0`
`:.(3-lamda)((-21-4lamda+lamda^2)-(-20))-10((-14+2lamda)-(-12))+5((-10)-(-9-3lamda))=0`
`:.(3-lamda)(-1-4lamda+lamda^2)-10(-2+2lamda)+5(-1+3lamda)=0`
`:. (-3-11lamda+7lamda^2-lamda^3)-(-20+20lamda)+(-5+15lamda)=0`
`:.(-lamda^3+7lamda^2-16lamda+12)=0`
`:.-(lamda-2)(lamda-2)(lamda-3)=0`
`:.(lamda-2)=0 or(lamda-2)=0 or(lamda-3)=0 `
`:.` The eigenvalues of the matrix `A` are given by `lamda=2,3`,
`:.` The eigenvalues of the matrix A are `2,2,3`
Here eigenvalue `2` is repeated, so matrix is derogatory matrix.
2. is Derogatory Matrix ?
`[[1,2,3],[4,5,6],[7,8,9]]`
Solution:
| `A` | = | | `1` | `2` | `3` | | | `4` | `5` | `6` | | | `7` | `8` | `9` | |
|
A matrix is derogatory if it's eigenvalues are repeated.
`|A-lamdaI|=0`
| `(1-lamda)` | `2` | `3` | | | `4` | `(5-lamda)` | `6` | | | `7` | `8` | `(9-lamda)` | |
| = 0 |
`:.(1-lamda)((5-lamda) × (9-lamda) - 6 × 8)-2(4 × (9-lamda) - 6 × 7)+3(4 × 8 - (5-lamda) × 7)=0`
`:.(1-lamda)((45-14lamda+lamda^2)-48)-2((36-4lamda)-42)+3(32-(35-7lamda))=0`
`:.(1-lamda)(-3-14lamda+lamda^2)-2(-6-4lamda)+3(-3+7lamda)=0`
`:. (-3-11lamda+15lamda^2-lamda^3)-(-12-8lamda)+(-9+21lamda)=0`
`:.(-lamda^3+15lamda^2+18lamda)=0`
`:.-lamda(lamda+1.11684397)(lamda-16.11684397)=0`
`:.lamda=0 or(lamda+1.11684397)=0 or(lamda-16.11684397)=0 `
`:.` The eigenvalues of the matrix `A` are given by `lamda=-1.11684397,0,16.11684397`,
`:.` The eigenvalues of the matrix A are `-1.1168,0,16.1168`
Here all eigenvalues are different, so matrix is non-derogatory matrix.
This material is intended as a summary. Use your textbook for detail explanation.
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