Determinant - Properties of The Determinant

Properties of The Determinant

The determinant has many properties. Some basic properties of determinants are:

1. where I_n is the n×n identity matrix.
2. For square matrices A and B of equal size,

5. for an n×n matrix.
6. If A is a triangular matrix, i.e. a_i,j = 0 whenever i > j or, alternatively, whenever i < j, then its determinant equals the product of the diagonal entries:

This can be deduced from some of the properties below, but it follows most easily directly from the Leibniz formula (or from the Laplace expansion), in which the identity permutation is the only one that gives a non-zero contribution.

A number of additional properties relate to the effects on the determinant of changing particular rows or columns:

7. Viewing an n×n matrix as being composed of n columns, the determinant is an n-linear function. This means that if one column of a matrix A is written as a sum v + w of two column vectors, and all other columns are left unchanged, then the determinant of A is the sum determinants of the matrices obtained from A by replacing the column by v respectively by w (and a similar relation holds when writing a column as a scalar multiple of a column vector).
8. This n-linear function is an alternating form. This means that whenever two columns of a matrix are identical, or more generally some column can be expressed as a linear combination of the other columns (i.e. the columns of the matrix form a linearly dependent set), its determinant is 0.

Properties 1, 7 and 8 — which all follow from the Leibniz formula — completely characterize the determinant; in other words the determinant is the unique function from n×n matrices to scalars that is n-linear alternating in the columns, and takes the value 1 for the identity matrix (this characterization holds even if scalars are taken in any given commutative ring). To see this it suffices to expand the determinant by multi-linearity in the columns into a (huge) linear combination of determinants of matrices in which each column is a standard basis vector. These determinants are either 0 (by property 8) or else ±1 (by properties 1 and 11 below), so the linear combination gives the expression above in terms of the Levi-Civita symbol. While less technical in appearance, this characterization cannot entirely replace the Leibniz formula in defining the determinant, since without it the existence of an appropriate function is not clear. For matrices over non-commutative rings, properties 7 and 8 are incompatible for n ≥ 2, so there is no good definition of the determinant in this setting.

Property 2 above implies that properties for columns have their counterparts in terms of rows:

9. Viewing an n×n matrix as being composed of n rows, the determinant is an n-linear function.
10. This n-linear function is an alternating form: whenever two rows of a matrix are identical, its determinant is 0.
11. Interchanging two columns of a matrix multiplies its determinant by −1. This follows from properties 7 and 8 (it is a general property of multilinear alternating maps). Iterating gives that more generally a permutation of the columns multiplies the determinant by the sign of the permutation. Similarly a permutation of the rows multiplies the determinant by the sign of the permutation.
12. Adding a scalar multiple of one column to another column does not change the value of the determinant. This is a consequence of properties 7 and 8: by property 7 the determinant changes by a multiple of the determinant of a matrix with two equal columns, which determinant is 0 by property 8. Similarly, adding a scalar multiple of one row to another row leaves the determinant unchanged.

These properties can be used to facilitate the computation of determinants by simplifying the matrix to the point where the determinant can be determined immediately. Specifically, for matrices with coefficients in a field, properties 11 and 12 can be used to transform any matrix into a triangular matrix, whose determinant is given by property 6; this is essentially the method of Gaussian elimination.

For example, the determinant of can be computed using the following matrices:

Here, B is obtained from A by adding −1/2 × the first row to the second, so that det(A) = det(B). C is obtained from B by adding the first to the third row, so that det(C) = det(B). Finally, D is obtained from C by exchanging the second and third row, so that det(D) = −det(C). The determinant of the (upper) triangular matrix D is the product of its entries on the main diagonal: (−2) · 2 · 4.5 = −18. Therefore det(A) = −det(D) = +18.