Refresher - Properties of the Determinants

Properties of the determinant

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

1. The determinant of the n×n identity matrix equals 1.
2. Viewing an n×n matrix as being composed of n columns, the determinant is an n-linearfunction. 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).
3. This n-linear function is an alternatingform. This means that whenever two columns of a matrix are identical,
   its determinant is 0.
4. A matrix and its transpose have the same determinant. This implies that properties for columns have their counterparts in terms of rows:
5. Viewing an n×n matrix as being composed of n rows, the determinant is an n-linear function.
6. This n-linear function is an alternating form: whenever two rows of a matrix are identical, its determinant is 0.
7. Interchanging two columns of a matrix multiplies its determinant by −1. This follows from properties 2 and 3 (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.
8. Adding a scalar multiple of one column to another column does not change the value of the determinant. This is a consequence of properties 2 and 3: by property 2 the determinant changes by a multiple of the determinant of a matrix with two equal columns, which determinant is 0 by property 3. Similarly, adding a scalar multiple of one row to another row leaves the determinant unchanged.
9. Let A be ann ×nmatrix and c be a scalar then,${\displaystyle det\left(cA\right)=c^{n}det\left(A\right)}$
10. If A and B are matrices of the same size then ${\displaystyle det\left(AB\right)=det\left(A\right)det\left(B\right)}$
11. Suppose that A is an invertible matrix then, ${\displaystyle det\left(A^{-1}\right)=\frac{1}{det\left(A\right)}}$
12. If A is a square matrix then, ${\displaystyle det\left(A\right)=det\left(A^{-1}\right)}$.
13. Suppose that A is an n×n triangular matrix then,${\displaystyle det\left(A\right)=a_{11}{a}_{22}\dots a_{\cap }}$.