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-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.

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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-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.

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