Introduction to Determinants

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.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det(AB) = \det(A)\det(B) \,} for all n-by-n matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} .

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

• It is easy to see that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det(rI_n) = r^n \,} and thus

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det(rA) = \det(rI_n \cdot A) = r^n \det(A) \,} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -by-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and all scalars Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det(A^\top) = \det(A). \,}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \det(A^*) = \det(A)^*. \,}