Determinant of a Matrix

The determinant of a matrix is a value associated with a matrix (or the vectors defining it), noted $ \det(M) $ or $ |M| $, this value is calculated from the coefficients of the matrix and is used in various matrix calculations.

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

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

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:

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

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.

An identity matrix $ I_n $ has for determinant $ 1 $ whatever the value of $ n $.

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

The Sarrus Rule is a practical method for manually calculating the determinant of a 3×3 matrix quickly by copying 2 columns:

The determinant of M is calculated as follows $ (aei+bfg+cdh) - (gec+hfa-idb) $, it is a subtraction of 2 sums. The first is composed of the multiplication of the elements of the three main diagonals (from top to bottom, from left to right) $ aei $, $ bfg $ and $ cdh $ and the second of the multiplication of the elements of the three secondary diagonals (from bottom to top, from left to right) $ gec $, $ hfa $ and $ idb $

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

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.)

A matrix M is invertible if and only if its determinant is non-zero.

If $ \det(M) \neq 0 $, then $ M $ is invertible.

If two rows (or columns) of a matrix are identical, then its determinant is zero (= 0).