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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=15-3/5xx25R_2 larr R_2-3/5xx R_1 9 9=18-3/5xx15R_2 larr R_2-3/5xx R_1 3 3=0-3/5xx-5R_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+1/5xx25R_3 larr R_3+1/5xx R_1 3 3=0+1/5xx15R_3 larr R_3+1/5xx R_1 10 10=11+1/5xx-5R_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=0-1/3xx0R_3 larr R_3-1/3xx R_2 0 0=3-1/3xx9R_3 larr R_3-1/3xx R_2 9 9=10-1/3xx3R_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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