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) : Mathematics, Algebra, Symbolic Computation

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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|.

For a 2x2 matrix, 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 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 idea is the same for higher matrix sizes:

For 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 formula easier than the explaination 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 as determinant of 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.

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