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,...,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:

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

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

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