Laplace's rule

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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 saidis the determinant obtained by eliminating the i-th row and the i-th column of Zmxn.

  • Mijis the i, j-th less. The determinant I saidin function of Mijis called the i, j-th cofactorof the matrix Zmxn.
  • to is the sign setting of the position.

Theoretical example of Laplace's rule

We define A3 × 3 as:

  1. 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 lessto 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.

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