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Determinant of a Matrix

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|.

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Determinant of a Matrix -

Tag(s) : Mathematics,Algebra,Symbolic Computation

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Determinant of a Matrix

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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|.

Answers to Questions

How to calculate a matrix determinant?

For a 2x2 matrix, the calculation is:

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

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

For higher size matrix like 3x3, one computes:$$ \begin{vmatrix} a & b & c\\d & e & f\\g & h & i \end{vmatrix} = a\begin{vmatrix} e & f\\h & i \end{vmatrix} - b\begin{vmatrix} d & f\\g & i \end{vmatrix} + c\begin{vmatrix} d & e\\g & h \end{vmatrix}\\= aei-afh+bfg-bdi+cdh-ceg $$The idea is the same for higher matrix sizes:$$ \begin{vmatrix} a & b & c & d\\e & f & g & h\\i & j & k & l\\m & n & o & p \end{vmatrix} =a\begin{vmatrix} f & g & h\\j & k & l\\n & o & p \end{vmatrix} -b\begin{vmatrix} e & g & h\\i & k & l\\m & o & p \end{vmatrix} +c\begin{vmatrix} e & f & h\\i & j & l\\m & n & p \end{vmatrix} -d\begin{vmatrix} e & f & g\\i & j & k\\m & n & o \end{vmatrix} $$

How to compute the determinant of a matrix 1x1?

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

What is the determinant of an identity matrix?

An identity matrix has as determinant of 1.

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