Find Determinants using montante (bareiss algorithm) ...
`[[1,2,3],[0,1,0],[2,3,1]]`

Solution:

`A`

=

`1` `2` `3` `0` `1` `0` `2` `3` `1`

Step-0: Previous Pivot `=1`

Step-1: Pivot `=A_(1,1)=1`, Previous Pivot `=1`

`A_(i,j)=(A_(1,1) * A_(i,j) - A_(i,1) * A_(1,j))/(1)`, where `i>1` and `j>1`

`A_(2,2)=(1 * 1 - 0 * 2)/(1)=1`

`A_(2,3)=(1 * 0 - 0 * 3)/(1)=0`

`A_(3,2)=(1 * 3 - 2 * 2)/(1)=-1`

`A_(3,3)=(1 * 1 - 2 * 3)/(1)=-5`

So matrix becomes

`A`

=

`1` `2` `3` `0` `1` `0` `2` `-1` `-5`

Step-2: Pivot `=A_(2,2)=1`, Previous Pivot `=1`

`A_(i,j)=(A_(2,2) * A_(i,j) - A_(i,2) * A_(2,j))/(1)`, where `i>2` and `j>2`

`A_(3,3)=(1 * -5 - -1 * 0)/(1)=-5`

So matrix becomes

`A`

=

`1` `2` `3` `0` `1` `0` `2` `-1` `-5`

Determinant will be `A_(3,3)` multiplied by the scaling factor 1

`:.` Determinant `=-5`

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