# 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.
$\det \begin{bmatrix} \ddots & \vdots & \ldots \\ \lambda a_1 + \mu b_1 & \cdots & \lambda a_n + \mu b_n \\ \cdots & \vdots & \ddots \end{bmatrix} = \lambda \det \begin{bmatrix} \ddots & \vdots & \cdots \\ a_1 & \cdots & a_n \\ \cdots & \vdots & \ddots \end{bmatrix} + \mu \det \begin{bmatrix} \ddots & \vdots & \cdots \\ b_1 & \cdots & b_n \\ \cdots & \vdots & \ddots \end{bmatrix}$
• 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:
$\det \begin{bmatrix} \cdots \\ \mbox{row A} \\ \cdots \\ \mbox{row B} \\ \cdots \end{bmatrix} = - \det \begin{bmatrix}\cdots \\ \mbox{row B} \\ \cdots \\ \mbox{row A} \\ \cdots \end{bmatrix}$

• 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 $\det(rI_n) = r^n \,$ and thus
$\det(rA) = \det(rI_n \cdot A) = r^n \det(A) \,$ 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,...,vn in Rn form a basis if and only if det(v1,...,vn) is non-zero.

A matrix and its transpose have the same determinant:

$\det(A^\top) = \det(A). \,$

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

$\det(A^*) = \det(A)^*. \,$

## Theorems

### Existance

Using Laplace's formula for the determinant

### Binet's theorem

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