#### Rules

#### Calculator

- Determinant 2x2
- Determinant 3x3
- Determinant 3x3 symbolic
- Determinant 4x4
- Determinant 4x4 symbolic
- Determinant 5x5
- Determinant NxN

### Laplace Expansion Theorem

The Laplace expansion theorem provides a method for calculating the determinant, wherein the determinant is developed according to a row or column. The dimension is reduced and can be gradually reduced more and more up to the scalar.

### Example of the Laplace expansion according to the first row on a 3x3 Matrix.

$\mathrm{det\; A}=\left|\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}\right|$

The first element is given by the factor a_{11} and the sub-determinant consisting of the elements with green background.

$\left|\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}\right|=>{a}_{11}\left|\begin{array}{cc}{a}_{22}& {a}_{23}\\ {a}_{32}& {a}_{33}\end{array}\right|$

The second element is given by the factor a_{12} and the sub-determinant consisting of the elements with green background.

$\left|\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}\right|=>{a}_{12}\left|\begin{array}{cc}{a}_{21}& {a}_{23}\\ {a}_{31}& {a}_{33}\end{array}\right|$

The third element is given by the factor a_{13} and the sub-determinant consisting of the elements with green background.

$\left|\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}\right|=>{a}_{13}\left|\begin{array}{cc}{a}_{21}& {a}_{22}\\ {a}_{31}& {a}_{32}\end{array}\right|$

With the three elements the determinant can be written as a sum of 2x2 determinants.

$\mathrm{det\; A}=\left|\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}\right|$

$={a}_{11}\left|\begin{array}{cc}{a}_{22}& {a}_{23}\\ {a}_{32}& {a}_{33}\end{array}\right|$

$-{a}_{12}\left|\begin{array}{cc}{a}_{21}& {a}_{23}\\ {a}_{31}& {a}_{33}\end{array}\right|$

$+{a}_{13}\left|\begin{array}{cc}{a}_{21}& {a}_{22}\\ {a}_{31}& {a}_{32}\end{array}\right|$

It is important to consider that the sign of the elements alternate in the following manner.

$\left|\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}\right|$

### Gauss Method

With the Gauss method, the determinant is so transformed that the elements of the lower triangle matrix become zero. To do this, you use the row-factor rules and the addition of rows. The addition of rows does not change the value of the determinate. Factors of a row must be considered as multipliers before the determinat. If the determinat is triangular and the main diagonal elements are equal to one, the factor before the determinant corresponds to the value of the determinant itself.