Tool to compute a matrix determinant. The determinant of a square matrix M is a useful value computed from its inner elements and denoted det(M) or |M|.

Determinant of a Matrix - dCode

Tag(s) : Matrix

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The determinant of a matrix is a value associated with a matrix (or with the vectors defining it), this value is very practical in various matrix calculations.

For a 2x2 square matrix (order 2), the calculation is:

$$ \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc $$

__Example:__ $$ \begin{vmatrix} 1 & 2 \\ 3 & 4 \end{vmatrix} = 1 \times 4 - 2 \times 3 = -2 $$

For higher size matrix like order 3 (3x3), compute:

$$ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a \begin{vmatrix} e & f \\ h & i \end{vmatrix} - b \begin{vmatrix} d & f \\ g & i \end{vmatrix} + c \begin{vmatrix} d & e \\ g & h \end{vmatrix} \\ = aei-afh+bfg-bdi+cdh-ceg $$

The calculated sub-matrices are called minors of the original matrix.

The idea is the same for higher matrix sizes:

For an order 4 determinant of a 4x4 matrix:

$$ \begin{vmatrix} a & b & c & d \\ e & f & g & h \\ i & j & k & l \\ m & n & o & p \end{vmatrix} = a \begin{vmatrix} f & g & h \\ j & k & l \\ n & o & p \end{vmatrix} - b \begin{vmatrix} e & g & h \\ i & k & l \\ m & o & p \end{vmatrix} + c \begin{vmatrix} e & f & h \\ i & j & l \\ m & n & p \end{vmatrix} - d \begin{vmatrix} e & f & g \\ i & j & k \\ m & n & o \end{vmatrix} \\ = \\ a(fkp − flo − gjp + gln + hjo − hkn) − b(ekp − elo − gip + glm + hio − hkm) + c(ejp − eln − fip + flm + hin − hjm) − d(ejo − ekn − fio + fkm + gin − gjm) \\ = \\ afkp − aflo − agjp + agln + ahjo − ahkn − bekp + belo + bgip − bglm − bhio + bhkm + cejp − celn − cfip + cflm + chin − chjm − dejo + dekn + dfio − dfkm − dgin + dgjm $$

The determinant of a non-square matrix is not defined, it does not exist according to the definition of the determinant.

There is no other formula than the explanation above for the general case of a matrix of order n.

For a 1x1 matrix, the determinant is the only item of the matrix.

__Example:__ $$ | 1 | = 1 $$

An identity matrix has for determinant $ 1 $.

__Example:__ $$ \begin{vmatrix} 1 & 0 \\ 0 & 1 \end{vmatrix} = 1 \times 1 - 0 \times 0 $$

__Example:__ $$ \begin{vmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix} = ( 1 \times 1 \times 1) - (1 \times 0 \times 0) + (0 \times 0 \times 0) - (0 \times 0 \times 1) + (0 \times 0 \times 0) - (0 \times 1 \times 0) = 1 $$

Only the term corresponding to the multiplication of the diagonal will be 1 and the other terms will be null.

A transpose matrix has the same determinant as the untransposed matrix and hence a matrix has the same determinant as its own transpose matrix.

The determinant of a matrix is the product of its eigenvalues (including complex values and potential multiplicity).

This property is valid for any size of square matrix (2x2, 3x3, 4x4, 5x5, etc.)

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*Determinant of a Matrix* on dCode.fr [online website], retrieved on 2022-12-09,

- Matrix 2x2 Determinant Calculator
- Matrix 3x3 Determinant Calculator
- Matrix 4x4 Determinant Calculator
- Matrix NxN Determinant Calculator
- What is the matrix determinant? (Definition)
- How to calculate a matrix determinant?
- How to calculate the determinant for a non square matrix?
- What is the formula for calculating the determinant of a matrix of order n?
- How to compute the determinant of a matrix 1x1?
- What is the determinant of an identity matrix?
- What is the determinant of a transpose matrix?
- How to find a matrix determinant from its eigenvalues?

determinant,matrix,det,square,identity

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