# Linear Algebra/Determinant

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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.

- 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 , making the definition of the determinant on the rows equal to the one on the columns.

## Properties[edit]

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

- The determinant is a
*multiplicative map*in the sense that

- for all
*n*-by-*n*matrices and .

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

- It is easy to see that and thus

- for all -by- matrices and all scalars .

- 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

Expressed differently: the vectors *v*_{1},...,*v*_{n} in **R**^{n} form a basis if and only if det(*v*_{1},...,*v*_{n}) is non-zero.

A matrix and its transpose have the same determinant:

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

## Theorems[edit]

### Uniqueness[edit]

### Existence[edit]

Using Laplace's formula for the determinant

### Binet's theorem[edit]