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Find Is Positive Definite matrix [[25,15,-5],[15,18,0],[-5,0,11]]

Solution:
Your problem `->` Is Positive Definite matrix [[25,15,-5],[15,18,0],[-5,0,11]]


A matrix is positive definite if it’s symmetric and all its pivots are positive.

`A` = 
`25``15``-5`
`15``18``0`
`-5``0``11`


Test method 1: Existence of all Positive Pivots.
First apply Gaussian Elimination method to find Pivots
`A` = 
`25``15``-5`
`15``18``0`
`-5``0``11`


`R_2 larr R_2-3/5xx R_1`

 = 
`25``15``-5`
 `0` `0=15-3/5xx25`
`R_2 larr R_2-3/5xx R_1`
 `9` `9=18-3/5xx15`
`R_2 larr R_2-3/5xx R_1`
 `3` `3=0-3/5xx-5`
`R_2 larr R_2-3/5xx R_1`
`-5``0``11`


`R_3 larr R_3+1/5xx R_1`

 = 
`25``15``-5`
`0``9``3`
 `0` `0=-5+1/5xx25`
`R_3 larr R_3+1/5xx R_1`
 `3` `3=0+1/5xx15`
`R_3 larr R_3+1/5xx R_1`
 `10` `10=11+1/5xx-5`
`R_3 larr R_3+1/5xx R_1`


`R_3 larr R_3-1/3xx R_2`

 = 
`25``15``-5`
`0``9``3`
 `0` `0=0-1/3xx0`
`R_3 larr R_3-1/3xx R_2`
 `0` `0=3-1/3xx9`
`R_3 larr R_3-1/3xx R_2`
 `9` `9=10-1/3xx3`
`R_3 larr R_3-1/3xx R_2`


Pivots are the first non-zero element in each row of this eliminated matrix.

`:.` Pivots are `25,9,9`

Here all pivots are positive, so matrix is positive definate.


Test method 2: Determinants of all upper-left sub-matrices are positive.
`A` = 
`25``15``-5`
`15``18``0`
`-5``0``11`


 `25` 
`=25`


 `25`  `15` 
 `15`  `18` 
`=225`


 `25`  `15`  `-5` 
 `15`  `18`  `0` 
 `-5`  `0`  `11` 
`=2025`


Dets are `25,225,2025`

Here all determinants are positive, so matrix is positive definate.





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