Linear Algebra/Determinant

The determinant is a function which associates to a square matrix an element of the field on which it is defined (commonly the real or complex numbers). The determinant is required to hold these properties:

• It is linear on the rows of the matrix.

$\det \left[\begin{array}{ccc}\ddots & ⋮& \dots \\ \lambda a_1+\mu b_1& \cdots & \lambda a_n+\mu b_n\\ \cdots & ⋮& \ddots \end{array}\right]=\lambda \det \left[\begin{array}{ccc}\ddots & ⋮& \cdots \\ a_1& \cdots & a_n\\ \cdots & ⋮& \ddots \end{array}\right]+\mu \det \left[\begin{array}{ccc}\ddots & ⋮& \cdots \\ b_1& \cdots & b_n\\ \cdots & ⋮& \ddots \end{array}\right]$

• If the matrix has two equal rows its determinant is zero.
• The determinant of the identity matrix is 1.

It is possible to prove that $\det A=\det A^T$, making the definition of the determinant on the rows equal to the one on the columns.

Properties

• The determinant is zero if and only if the rows are linearly dependent.
• Changing two rows changes the sign of the determinant:

$\det \left[\begin{array}{c}\cdots \\ \text{row A}\\ \cdots \\ \text{row B}\\ \cdots \end{array}\right]=-\det \left[\begin{array}{c}\cdots \\ \text{row B}\\ \cdots \\ \text{row A}\\ \cdots \end{array}\right]$

• The determinant is a multiplicative map in the sense that

$\det (AB)=\det (A)\det (B)$ for all n-by-n matrices $A$ and $B$.

This is generalized by the Cauchy-Binet formula to products of non-square matrices.

• It is easy to see that $\det (r{I}_n)=r^n$ and thus

$\det (rA)=\det (r{I}_n\cdot A)=r^n\det (A)$ for all $n$-by-$n$ matrices $A$ and all scalars $r$.

• A matrix over a commutative ring R is invertible if and only if its determinant is a unit in R. In particular, if A is a matrix over a field such as the real or complex numbers, then A is invertible if and only if det(A) is not zero. In this case we have

$\det (A^{-1})=\det (A{)}^{-1}.$

Expressed differently: the vectors v₁,...,v_n in Rⁿ form a basis if and only if det(v₁,...,v_n) is non-zero.

A matrix and its transpose have the same determinant:

$\det (A^{\mathrm{\top }})=\det (A).$

The determinants of a complex matrix and of its conjugate transpose are conjugate:

$\det (A^{*})=\det (A{)}^{*}.$

Theorems

Existence

Using Laplace's formula for the determinant

Binet's theorem

$\det (AB)=\det A\cdot \det B$