# Laplace's rule

Laplace's rule is a method that allows you to quickly calculate the determinant of a square matrix with dimension 3 × 3 or greater by means of a recursive expansion series.

In other words, Laplace's rule factors the initial matrix into lower-dimensional matrices and adjusts its sign based on the position of the element in the matrix.

This method can be performed using rows or columns.

Recommended articles: matrices, matrix typologies and determinant of a matrix.

### Laplace's rule formula

Given any matrix Zmxn of dimension *mxn,*where m = n, it expands with respect to the i-th row, then:

*I said*is the determinant obtained by eliminating the i-th row and the i-th column of Zmxn.

*Mij*is the i, j-th*less*. The determinant*I said*in function of*Mij*is called the i, j-th*cofactor*of the matrix Zmxn.*to*is the sign setting of the position.

## Theoretical example of Laplace's rule

We define A3 × 3 as:

- Let's start with the first element a11. We grate the rows and columns that make up a11. The elements that remain without grating, will be the first determinant
*less*multiplied by a11.

2. We continue with the second element of the first row, that is, a12. We repeat the process: we grate the rows and columns that contain a12.

We adjust the sign of the minor:

We add the second determinant *less*to the previous result and we form an expansion series such that:

3. We continue with the third element of the first row, that is, a13. We repeat the process: we scratch the row and column that contain a13.

We add the third determinant *less* to the previous result and we extend the expansion series such that:

Since there are no more elements left in the first row, then we close the recursive process. We calculate the determinants *minors*.

In the same way that elements from the first row have been used, this method can also be applied with columns.

## Laplace's rule practical example

We define A3 × 3 as:

1. Let's start with the first element r11 = 5. We grate the rows and columns that make up a11 = 5. The elements that remain without grating, will be the first determinant *less* multiplied by a11 = 5.

2. We continue with the second element of the first row, that is, r12 = 2. We repeat the process: we grate the rows and columns that contain r12 = 2.

We adjust the sign of the minor:

We add the second determinant *less* to the previous result and we form an expansion series such that:

3. We continue with the third element of the first row, that is, r13 = 3. We repeat the process: we scratch the row and column that contain r13 = 3.

We add the third determinant *less* to the previous result and we extend the expansion series such that:

The determinant of the R3 × 3 matrix is 15.