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

Existence
Using Laplace's formula for the determinant

Binet's theorem
$$\det(A B) = \det A \cdot \det B $$