Determinant of a matrix
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There are exercises as selftest for this article. 
The determinant can be computed for an nbyn square matrix. In the context of matrices the determinant of a matrix is a special function that assigns a scalar value to the matrix. It is denoted with or in matrix structure with vertical lines:
Considering a matrix as a linear system the determinant provides information about its solvability. If the determinant is nonzero the linear system is clearly solvable. This feature is also important for matrix inversion.
Properties
The determinant is a multiplicative mapping, such that
for all nbyn square matrices and .
It holds for all nbyn square matrices and all scalar values :
A matrix and its transposed have the same determinant:
The determinant of an identity matrix is always 1:
2by2 matrices
For a 2by2 matrix the determinant can easily computed as follows:
Example: Determinant of a 2by2 matrix

3by3 matrices
For 3by3 matrices there is a formula called Rule of Sarrus to compute the determinant. The first two columns of the matrix are noted anew on the right of matrix. Then six diagonals are formed that describe the arithmetic operations to be performed:
The sum of the diagonal elements then forms the determinant:
The Khan Academy^{[1]} provides a good video ^{[2]} where this formula is explained. Please watch the video for further information about the computation of the determinant of a 3by3 matrix.
Example: Determinant of a 3by3 matrix

4by4 matrices
One possibility to compute the determinant of a 4by4 matrix is a formula that uses the minors and cofactors of a matrix. Please read the article about minors and cofactors before you continue with this article. First one row or column has to be choosen. The sum of the four corresponding values of the row or column multiplied by the related cofactors results in the determinant. If a column is choosen, it holds:
And if a row is choosen:
Example: determinant of a 4by4 matrix
This example describes the computation of the determinant of the transformation matrix that is introduced in the robotics script in chapter 3 on page 337 and used on the following pages.
It is always useful to choose a row or column with many zero values so that the corresponding products are omitted and the cofactors have not to be computed. For the matrix it is useful to choose row 4 because it contains three zero values and a one as factors: 
References
 ↑ https://www.khanacademy.org/ Khan Academy
 ↑ https://www.khanacademy.org/.../findingthedeterminantofa3x3matrixmethod1 Determinant of a 3by3 matrix