| # Determinant of a square matrix Definition An important result Geometry Formula Determinant and inverse Some properties We review a few important facts abut determinants here. For a self-contained exposition that includes proofs, see this text by Carl de Boor. ## Definition The determinant of a square, matrix , denoted , is defined by an algebraic formula of the coefficients of . The following formula for the determinant, known as Laplace's expansion formula, allows to compute the determinant recursively: where is the matrix obtained from by removing the -th row and first column. (The first column does not play a special role here: the determinant remains the same if we use any other column.) The determinant is the unique function of the entries of such that . is a linear function of any column (when the others are fixed). changes sign when two columns are permuted. There are other expressions of the determinant, including Leibnitz formula (proven here): where denotes the set of permutations of the integers . Here, denotes the sign of the permutation , which is the number of pairwise exchanges required to transform into . ## Important result An important result is that a square matrix is invertible if and only if its determinant is not zero. We use this key result when introducing eigenvalues of symmetric matrices. ## Geometry
| The determinant of a matrix with columns is the volume of the parallepiped defined by the vectors . (Source: wikipedia). Hence the determinant is a measure of scale that quantifies how the linear map associated with , , changes volumes.
In general, the absolute value of the determinant of a matrix is the volume of the parallepiped This is consistent with the fact that when is not invertible, its columns define a parallepiped of zero volume. ## Determinant and inverse The determinant can be used to compute the inverse of a square, full rank (that is, invertible) matrix : the inverse has elements given by , where is a matrix obtained from by removing its -th row and -th column. For example, the determinant of a matrix is given by It is indeed the volume of the area of a parallepiped defined with the columns of , . The inverse is given by ## Some properties ### Determinant of triangular matrices If a matrix is square, triangular, then its determinant is simply the product of its diagonal coefficients. This comes right from Laplace's expansion formula above. ### Determinant of transpose The determinant of a square matrix and that of its transpose are equal. ### Determinant of a product of matrices For two invertible square matrices, we have In particular: This also implies that for an orthogonal matrix , that is, a matrix with , we have ### Determinant of block matrices As a generalization of the above result, we have for three compatible blocks : A more general formula is