1. Find Cholesky Decomposition ...
`[[6,15,55],[15,55,225],[55,225,979]]`

Solution:

Here matrix is symmetric positive definite, so Cholesky decomposition is possible.

`A`

=

`6` `15` `55` `15` `55` `225` `55` `225` `979`

A matrix is positive definite if Determinants of all upper-left sub-matrices are positive.

Test method 2: Determinants of all upper-left sub-matrices are positive.

`A`

=

`6` `15` `55` `15` `55` `225` `55` `225` `979`

`6`

`=6`

`6`   `15`   `15`   `55`

`=105`

`6`   `15`   `55`   `15`   `55`   `225`   `55`   `225`   `979`

`=3920`

Determinants are `6,105,3920`

Here all determinants are positive, so matrix is positive semi-definite.

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Formula
`l_(ki)=(a_(ki) - sum_{j=1}^{i-1} l_(ij) * l_(kj))/(l_(ii))`

`l_(kk)=sqrt(a_(kk)-sum_{j=1}^{k-1} l_(kj)^2)`

Here `A`

=

6 15 55 15 55 225 55 225 979

`l_(11)=sqrt(a_(11))=sqrt(6)=2.4495`

`l_(21)=(a_(21))/l_(11)=(15)/(2.4495)=6.1237`

`l_(22)=sqrt(a_(22)-l_(21)^2)=sqrt(55-(6.1237)^2)=sqrt(55-37.5)=4.1833`

`l_(31)=(a_(31))/l_(11)=(55)/(2.4495)=22.4537`

`l_(32)=(a_(32)-l_(31) xx l_(21))/l_(22)=(225-(22.4537)xx(6.1237))/(4.1833)=(225-137.5)/(4.1833)=20.9165`

`l_(33)=sqrt(a_(33)-l_(31)^2-l_(32)^2)=sqrt(979-(22.4537)^2-(20.9165)^2)=sqrt(979-941.6667)=6.1101`

So `L`

=

`l_(11)` `0` `0` `l_(21)` `l_(22)` `0` `l_(31)` `l_(32)` `l_(33)`

=

2.4495 0 0 6.1237 4.1833 0 22.4537 20.9165 6.1101

`L xx L^T`

=

2.4495 0 0 6.1237 4.1833 0 22.4537 20.9165 6.1101

`xx`

2.4495 6.1237 22.4537 0 4.1833 20.9165 0 0 6.1101

=

6 15 55 15 55 225 55 225 979

and `A`

=

6 15 55 15 55 225 55 225 979

Solution is possible.