# §1.3 Determinants

## §1.3(i) Definitions and Elementary Properties

 1.3.1 $\det[a_{jk}]=\begin{vmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{vmatrix}=a_{11}a_{22}-a_{12}a_{21}.$ ⓘ Symbols: $\det$: determinant, $j$: integer and $k$: integer Permalink: http://dlmf.nist.gov/1.3.E1 Encodings: TeX, pMML, png See also: Annotations for §1.3(i), §1.3 and Ch.1
 1.3.2 $\det[a_{jk}]=\begin{vmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{vmatrix}=a_{11}\begin{vmatrix}a_{22}&a_{23}\\ a_{32}&a_{33}\end{vmatrix}-a_{12}\begin{vmatrix}a_{21}&a_{23}\\ a_{31}&a_{33}\end{vmatrix}+a_{13}\begin{vmatrix}a_{21}&a_{22}\\ a_{31}&a_{32}\end{vmatrix}=a_{11}a_{22}a_{33}-a_{11}a_{23}a_{32}-a_{12}a_{21}a% _{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{13}a_{22}a_{31}.$ ⓘ Symbols: $\det$: determinant, $j$: integer and $k$: integer Permalink: http://dlmf.nist.gov/1.3.E2 Encodings: TeX, pMML, png See also: Annotations for §1.3(i), §1.3 and Ch.1

Higher-order determinants are natural generalizations. The minor $M_{jk}$ of the entry $a_{jk}$ in the $n$th-order determinant $\det[a_{jk}]$ is the ($n-1$)th-order determinant derived from $\det[a_{jk}]$ by deleting the $j$th row and the $k$th column. The cofactor $A_{jk}$ of $a_{jk}$ is

 1.3.3 $A_{jk}=(-1)^{j+k}M_{jk}.$ ⓘ Defines: $A_{jk}$: cofactor (locally) Symbols: $j$: integer, $k$: integer and $M_{jk}$: minor Permalink: http://dlmf.nist.gov/1.3.E3 Encodings: TeX, pMML, png See also: Annotations for §1.3(i), §1.3 and Ch.1

An $n$th-order determinant expanded by its $j$th row is given by

 1.3.4 $\det[a_{jk}]=\sum^{n}_{\ell=1}a_{j\ell}A_{j\ell}.$ ⓘ Symbols: $\det$: determinant, $j$: integer, $k$: integer, $n$: nonnegative integer and $A_{jk}$: cofactor Permalink: http://dlmf.nist.gov/1.3.E4 Encodings: TeX, pMML, png See also: Annotations for §1.3(i), §1.3 and Ch.1

If two rows (or columns) of a determinant are interchanged, then the determinant changes sign. If two rows (columns) of a determinant are identical, then the determinant is zero. If all the elements of a row (column) of a determinant are multiplied by an arbitrary factor $\mu$, then the result is a determinant which is $\mu$ times the original. If $\mu$ times a row (column) of a determinant is added to another row (column), then the value of the determinant is unchanged.

 1.3.5 $\displaystyle\det[a_{jk}]^{\mathrm{T}}$ $\displaystyle=\det[a_{jk}],$ ⓘ Symbols: $\det$: determinant, $j$: integer and $k$: integer Permalink: http://dlmf.nist.gov/1.3.E5 Encodings: TeX, pMML, png See also: Annotations for §1.3(i), §1.3 and Ch.1 1.3.6 $\displaystyle\det[a_{jk}]^{-1}$ $\displaystyle=\frac{1}{\det[a_{jk}]},$ ⓘ Symbols: $\det$: determinant, $j$: integer and $k$: integer Permalink: http://dlmf.nist.gov/1.3.E6 Encodings: TeX, pMML, png See also: Annotations for §1.3(i), §1.3 and Ch.1 1.3.7 $\displaystyle\det([a_{jk}][b_{jk}])$ $\displaystyle=(\det[a_{jk}])(\det[b_{jk}]).$ ⓘ Symbols: $\det$: determinant, $j$: integer and $k$: integer Referenced by: §1.3(i) Permalink: http://dlmf.nist.gov/1.3.E7 Encodings: TeX, pMML, png See also: Annotations for §1.3(i), §1.3 and Ch.1

For real-valued $a_{jk}$,

 1.3.8 ${\begin{vmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{vmatrix}}^{2}\leq(a^{2}_{11}+a^{2}_{12})(a^{2}_{21}+a^{2}_{2% 2}),$ ⓘ Symbols: $\det$: determinant Permalink: http://dlmf.nist.gov/1.3.E8 Encodings: TeX, pMML, png See also: Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1
 1.3.9 $\det[a_{jk}]^{2}\leq\left(\sum^{n}_{k=1}a^{2}_{1k}\right)\left(\sum^{n}_{k=1}a% ^{2}_{2k}\right)\dots\left(\sum^{n}_{k=1}a^{2}_{nk}\right).$ ⓘ Symbols: $\det$: determinant, $j$: integer, $k$: integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.3.E9 Encodings: TeX, pMML, png See also: Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

Compare also (1.3.7) for the left-hand side. Equality holds iff

 1.3.10 $a_{j1}a_{k1}+a_{j2}a_{k2}+\dots+a_{jn}a_{kn}=0$ ⓘ Symbols: $j$: integer, $k$: integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.3.E10 Encodings: TeX, pMML, png See also: Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

for every distinct pair of $j,k$, or when one of the factors $\sum^{n}_{k=1}a^{2}_{jk}$ vanishes.

## §1.3(ii) Special Determinants

An alternant is a determinant function of $n$ variables which changes sign when two of the variables are interchanged. Examples:

 1.3.11 $\det[f_{k}(x_{j})],$ $j=1,\dots,n$; $k=1,\dots,n$, ⓘ Symbols: $\det$: determinant, $j$: integer, $k$: integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.3.E11 Encodings: TeX, pMML, png See also: Annotations for §1.3(ii), §1.3 and Ch.1 1.3.12 $\det[f(x_{j},y_{k})],$ $j=1,\dots,n$; $k=1,\dots,n$. ⓘ Symbols: $\det$: determinant, $j$: integer, $k$: integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.3.E12 Encodings: TeX, pMML, png See also: Annotations for §1.3(ii), §1.3 and Ch.1

### Vandermonde Determinant or Vandermondian

 1.3.13 $\begin{vmatrix}1&x_{1}&x^{2}_{1}&\cdots&x^{n-1}_{1}\\ 1&x_{2}&x^{2}_{2}&\cdots&x^{n-1}_{2}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 1&x_{n}&x^{2}_{n}&\cdots&x_{n}^{n-1}\end{vmatrix}=\prod_{1\leq j ⓘ Symbols: $\det$: determinant, $j$: integer, $k$: integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.3.E13 Encodings: TeX, pMML, png See also: Annotations for §1.3(ii), §1.3(ii), §1.3 and Ch.1

### Cauchy Determinant

 1.3.14 $\det\left[\frac{1}{a_{j}-b_{k}}\right]=(-1)^{n(n-1)/2}\*\prod_{1\leq j ⓘ Symbols: $\det$: determinant, $j$: integer, $k$: integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.3.E14 Encodings: TeX, pMML, png See also: Annotations for §1.3(ii), §1.3(ii), §1.3 and Ch.1

### Circulant

 1.3.15 $\begin{vmatrix}a_{1}&a_{2}&\cdots&a_{n}\\ a_{n}&a_{1}&\cdots&a_{n-1}\\ \vdots&\vdots&\ddots&\vdots\\ a_{2}&a_{3}&\cdots&a_{1}\end{vmatrix}=\prod^{n}_{k=1}(a_{1}+a_{2}\omega_{k}+a_% {3}\omega_{k}^{2}+\dots+a_{n}\omega_{k}^{n-1}),$ ⓘ Symbols: $\det$: determinant, $k$: integer, $n$: nonnegative integer and $\omega_{j}$: roots of unity Permalink: http://dlmf.nist.gov/1.3.E15 Encodings: TeX, pMML, png See also: Annotations for §1.3(ii), §1.3(ii), §1.3 and Ch.1

where $\omega_{1},\omega_{2},\dots,\omega_{n}$ are the $n$th roots of unity (1.11.21).

### Krattenthaler’s Formula

For

 1.3.16 $t_{jk}=(x_{j}+a_{n})(x_{j}+a_{n-1})\cdots(x_{j}+a_{k+1})\*(x_{j}+b_{k})(x_{j}+% b_{k-1})\cdots(x_{j}+b_{2}),$ ⓘ Defines: $t_{jk}$: elements (locally) Symbols: $j$: integer, $k$: integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.3.E16 Encodings: TeX, pMML, png See also: Annotations for §1.3(ii), §1.3(ii), §1.3 and Ch.1
 1.3.17 $\det[t_{jk}]=\prod_{1\leq j ⓘ Symbols: $\det$: determinant, $j$: integer, $k$: integer, $n$: nonnegative integer and $t_{jk}$: elements Referenced by: §1.3(ii) Permalink: http://dlmf.nist.gov/1.3.E17 Encodings: TeX, pMML, png See also: Annotations for §1.3(ii), §1.3(ii), §1.3 and Ch.1

## §1.3(iii) Infinite Determinants

Let $a_{j,k}$ be defined for all integer values of $j$ and $k$, and $D_{n}[a_{j,k}]$ denote the $(2n+1)\times(2n+1)$ determinant

 1.3.18 $D_{n}[a_{j,k}]=\begin{vmatrix}a_{-n,-n}&a_{-n,-n+1}&\dots&a_{-n,n}\\ a_{-n+1,-n}&a_{-n+1,-n+1}&\dots&a_{-n+1,n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n,-n}&a_{n,-n+1}&\dots&a_{n,n}\end{vmatrix}.$ ⓘ Defines: $D_{n}[a_{j,k}]$: $(2n+1)\times(2n+1)$ determinant (locally) Symbols: $\det$: determinant, $j$: integer, $k$: integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.3.E18 Encodings: TeX, pMML, png See also: Annotations for §1.3(iii), §1.3 and Ch.1

If $D_{n}[a_{j,k}]$ tends to a limit $L$ as $n\to\infty$, then we say that the infinite determinant $D_{\infty}[a_{j,k}]$ converges and $D_{\infty}[a_{j,k}]=L$.

Of importance for special functions are infinite determinants of Hill’s type. These have the property that the double series

 1.3.19 $\sum^{\infty}_{j,k=-\infty}|a_{j,k}-\delta_{j,k}|$ ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$: Kronecker delta, $j$: integer and $k$: integer Permalink: http://dlmf.nist.gov/1.3.E19 Encodings: TeX, pMML, png See also: Annotations for §1.3(iii), §1.3 and Ch.1

converges (§1.9(vii)). Here $\delta_{j,k}$ is the Kronecker delta. Hill-type determinants always converge.

For further information see Whittaker and Watson (1927, pp. 36–40) and Magnus and Winkler (1966, §2.3).