Determinant

Determinants are mathematical objects which are very useful in the analysis and solution of systems of linear equations. As shown in Cramer's Rule, a nonhomogeneous system of linear equations has a nontrivial solution Iff the determinant of the system's Matrix is Nonzero (so that the Matrix is nonsingular). A $2\times 2$ determinant is defined to be

$$\mathop{\rm det}\nolimits \left[{\matrix{a & b\cr c & d\cr}}... ... \left\vert\matrix{a & b\cr c & d\cr}\right\vert \equiv ad-bc.$$ (1)

A $k\times k$ determinant can be expanded by Minors to obtain

$\left\vert\matrix{a_{11} & a_{12} & a_{13} & \cdots & a_{1k}\cr a_{21} & a_{22}... ... & \vdots & \ddots & \vdots\cr a_{k2} & a_{k3} & \cdots & a_{kk}\cr}\right\vert$
$- a_{12}\left\vert\matrix{a_{21} & a_{23} & \cdots & a_{2k}\cr \vdots & \vdots... ...& \ddots & \vdots\cr a_{k1} & a_{k2} & \cdots & a_{k(k-1)}\cr}\right\vert.\quad$	(2)

A general determinant for a Matrix A has a value

$$\vert{\hbox{\sf A}}\vert = \sum_i a_{ij}a^{ij},$$ (3)

with no implied summation over $j$ and where $a^{ij}$ is the Cofactor of $a_{ij}$ defined by

$$a^{ij} \equiv (-1)^{i+j} C_{ij}.$$ (4)

Here, C is the $(n-1)\times (n-1)$ Matrix formed by eliminating row $i$ and column $j$ from A, i.e., by Determinant Expansion by Minors.

Given an $n\times n$ determinant, the additive inverse is

$$\vert-{\hbox{\sf A}}\vert = (-1)^n\vert{\hbox{\sf A}}\vert.$$ (5)

Determinants are also Distributive, so

$$\vert{\hbox{\sf A}}{\hbox{\sf B}}\vert = \vert{\hbox{\sf A}}\vert\,\vert{\hbox{\sf B}}\vert.$$ (6)

This means that the determinant of a Matrix Inverse can be found as follows:

$$\vert{\hbox{\sf I}}\vert=\vert{\hbox{\sf A}}{\hbox{\sf A}}^{... ...= \vert{\hbox{\sf A}}\vert\,\vert{\hbox{\sf A}}^{-1}\vert = 1,$$ (7)

where I is the Identity Matrix, so

$$\vert{\hbox{\sf A}}\vert={1\over \vert{\hbox{\sf A}}^{-1}\vert}.$$ (8)

Determinants are Multilinear in rows and columns, since

$$\left\vert\matrix{a_1 & a_2 & a_3\cr a_4 & a_5 & a_6\cr a_7 ... ...& 0 & a_3\cr a_4 & a_5 & a_6\cr a_7 & a_8 & a_9\cr}\right\vert$$ (9)

and

$$\left\vert\matrix{a_1 & a_2 & a_3\cr a_4 & a_5 & a_6\cr a_7 ... ...0 & a_2 & a_3\cr 0 & a_5 & a_6\cr a_7 & a_8 & a_9}\right\vert.$$ (10)

The determinant of the Similarity Transformation of a matrix is equal to the determinant of the original Matrix

$$\vert{\hbox{\sf B}}{\hbox{\sf A}}{\hbox{\sf B}}^{-1}\vert = ... ... {1\over \vert{\hbox{\sf B}}\vert} = \vert{\hbox{\sf A}}\vert.$$ (11)

The determinant of a similarity transformation minus a multiple of the unit Matrix is given by

$$\vert{\hbox{\sf B}}^{-1}{\hbox{\sf A}}{\hbox{\sf B}}-\lambda {\hbox{\sf I}}\vert = \vert{\hbox{\sf B}}^{-1}{\hbox{\sf A}}{\hbox{\sf B}}-{\hbox{\sf B... ...ert{\hbox{\sf B}}^{-1}({\hbox{\sf A}}-\lambda{\hbox{\sf I}}){\hbox{\sf B}}\vert$$

$$= \vert{\hbox{\sf B}}^{-1}\vert \,\vert{\hbox{\sf A}}-\lambda{\hbox... ...ert\, \vert{\hbox{\sf B}}\vert= \vert{\hbox{\sf A}}-\lambda{\hbox{\sf I}}\vert.$$ (12)

The determinant of a Matrix Transpose equals the determinant of the original Matrix,

$$\vert{\hbox{\sf A}}\vert = \vert{\hbox{\sf A}}^{\rm T}\vert,$$ (13)

and the determinant of a Complex Conjugate is equal to the Complex Conjugate of the determinant

$$\vert{\hbox{\sf A}}^*\vert = \vert{\hbox{\sf A}}\vert^*.$$ (14)

Let $\epsilon$ be a small number. Then

$$\vert{\hbox{\sf I}}+\epsilon{\hbox{\sf A}}\vert = 1+\epsilon... ...op{\rm Tr}\nolimits ({\hbox{\sf A}})+{\mathcal O}(\epsilon^2),$$ (15)

where $\mathop{\rm Tr}\nolimits ({\hbox{\sf A}})$ is the Trace of A. The determinant takes on a particularly simple form for a Triangular Matrix

$$\left\vert\matrix{ a_{11} & a_{21} & \cdots & a_{k1}\cr 0 ... ... & 0 & \cdots & a_{kk}\cr}\right\vert = \prod_{n=1}^k a_{nn}.$$ (16)

Important properties of the determinant include the following.

1. Switching two rows or columns changes the sign.

2. Scalars can be factored out from rows and columns.

3. Multiples of rows and columns can be added together without changing the determinant's value.

4. Scalar multiplication of a row by a constant $c$ multiplies the determinant by $c$.

5. A determinant with a row or column of zeros has value 0.

6. Any determinant with two rows or columns equal has value 0.

Property 1 can be established by induction. For a $2\times 2$ Matrix, the determinant is

$$\left\vert\begin{array}{cc}a_1 & b_1\\ a_2 & b_2\end{array}\right\vert = a_1b_2-b_1a_2 = -(b_1a_2-a_1b_2)$$

$$= -\left\vert\begin{array}{cc}b_1 & a_1\\ b_2 & a_2\end{array}\right\vert$$ (17)

For a $3\times 3$ Matrix, the determinant is

$\left\vert\matrix{a_1 & b_1 & c_1\cr a_2 & b_2 & c_2\cr a_3 & b_3 & c_3\cr}\rig... ...& c_3\cr}\right\vert+c_1\left\vert\matrix{a_2 & b_2\cr a_3 & b_3\cr}\right\vert$
$\quad = -\left({a_1\left\vert\matrix{c_2 & b_2\cr c_3 & b_3\cr}\right\vert+b_1\... ...ert\matrix{a_1 & c_1 & b_1\cr a_2 & c_2 & b_2\cr a_3 & c_3 & b_3\cr}\right\vert$
$\quad = -\left({-a_1\left\vert\matrix{b_2 & c_2\cr b_3 & c_3\cr}\right\vert+b_1... ...ert\matrix{b_1 & a_1 & c_1\cr b_2 & a_2 & c_2\cr b_3 & a_3 & c_3\cr}\right\vert$
$\quad = -\left({-a_1\left\vert\matrix{c_2 & b_2\cr c_3 & b_3\cr}\right\vert-b_1... ...rt\matrix{c_1 & b_1 & a_1\cr c_2 & b_2 & a_2\cr c_3 & b_3 & a_3\cr}\right\vert.$	(18)

Property 2 follows likewise. For $2\times 2$ and $3\times 3$ matrices,

$$\left\vert\matrix{ka_1 & b_1\cr ka_2 & b_2\cr}\right\vert = ... ..._2) = k\left\vert\matrix{a_1 & b_1\cr a_2 & b_2\cr}\right\vert$$ (19)

and

$$\left\vert\matrix{ka_1 & b_1 & c_1\cr ka_2 & b_2 & c_2\cr ka... ..._1 & c_1\cr a_2 & b_2 & c_2\cr a_3 & b_3 & c_3\cr}\right\vert.$$ (20)

Property 3 follows from the identity

$\left\vert\matrix{a_1+kb_1 & b_1 & c_1\cr a_2+kb_2 & b_2 & c_2\cr a_3+kb_3 & b_3 & c_3\cr}\right\vert$
$= (a_1+kb_1)\left\vert\matrix{b_2 & c_2\cr b_3 & c_3\cr}\right\vert-b_1\left\v... ...ert+ c_1\left\vert\matrix{a_2+kb_2 & b_2\cr a_3+kb_3 & b_3\cr}\right\vert.\quad$	(21)

If $a_{ij}$ is an $n\times n$ Matrix with $a_{ij}$ Real Numbers, then $\mathop{\rm det}[a_{ij}]$ has the interpretation as the oriented $n$-dimensional Content of the Parallelepiped spanned by the column vectors $[a_{i,1}]$, ..., $[a_{i,n}]$ in $\Bbb{R}^n$. Here, ``oriented'' means that, up to a change of $+$ or $-$ Sign, the number is the $n$-dimensional Content, but the Sign depends on the ``orientation'' of the column vectors involved. If they agree with the standard orientation, there is a $+$ Sign; if not, there is a $-$ Sign. The Parallelepiped spanned by the $n$-D vectors ${\bf v}_1$ through ${\bf v}_i$ is the collection of points

$$t_1 {\bf v}_1 + \ldots + t_i {\bf v}_i,$$ (22)

where $t_j$ is a Real Number in the Closed Interval [0,1].

There are an infinite number of $3\times 3$ determinants with no 0 or $\pm 1$ entries having unity determinant. One parametric family is

$$\left\vert\matrix{ -8n^2-8n & 2n+1 & 4n\cr -4n^2-4n & n+1 & 2n+1\cr -4n^2-4n-1 & n & 2n-1\cr}\right\vert.$$ (23)

Specific examples having small entries include

$$\left\vert\matrix{2 & 3 & 2\cr 4 & 2 & 3\cr 9 & 6 & 7\cr}\ri... ...x{2 & 3 & 6\cr 3 & 2 & 3\cr 17 & 11 & 16\cr}\right\vert, \dots$$ (24)

(Guy 1989, 1994).

See also Circulant Determinant, Cofactor, Hessian Determinant, Hyperdeterminant, Immanant, Jacobian, Knot Determinant, Matrix, Minor, Permanent, Vandermonde Determinant, Wronskian

References

Arfken, G. ``Determinants.'' §4.1 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 168-176, 1985.

Guy, R. K. ``Unsolved Problems Come of Age.'' Amer. Math. Monthly 96, 903-909, 1989.

Guy, R. K. ``A Determinant of Value One.'' §F28 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 265-266, 1994.