Determinants

In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It allows characterizing some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one). The determinant of a matrix A is denoted det(A), det A, or |A|.

In the case of a 2 × 2 matrix the determinant can be defined as

${\begin{aligned}|A|={\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc.\end{aligned}}$

Similarly, for a 3 × 3 matrix A, its determinant is

${\begin{aligned}|A|={\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}&=a\,{\begin{vmatrix}e&f\\h&i\end{vmatrix}}-b\,{\begin{vmatrix}d&f\\g&i\end{vmatrix}}+c\,{\begin{vmatrix}d&e\\g&h\end{vmatrix}}\\[3pt]&=aei+bfg+cdh-ceg-bdi-afh.\end{aligned}}$

Each determinant of a 2 × 2 matrix in this equation is called a minor of the matrix A. This procedure can be extended to give a recursive definition for the determinant of an n × n matrix, known as Laplace expansion.

Determinants occur throughout mathematics. For example, a matrix is often used to represent the coefficients in a system of linear equations, and determinants can be used to solve these equations (Cramer's rule), although other methods of solution are computationally much more efficient. Determinants are used for defining the characteristic polynomial of a matrix, whose roots are the eigenvalues. In geometry, the signed n-dimensional volume of a n-dimensional parallelepiped is expressed by a determinant. This is used in calculus with exterior differential forms and the Jacobian determinant, in particular for changes of variables in multiple integrals.

2 × 2 matrices

The determinant of a 2 × 2 matrix ${\begin{pmatrix}a&b\\c&d\end{pmatrix}}$ is denoted either by "det" or by vertical bars around the matrix, and is defined as

$\operatorname {det} {\begin{pmatrix}a&b\\c&d\end{pmatrix}}={\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc.$

For example,

$\operatorname {det} {\begin{pmatrix}3&7\\1&-4\end{pmatrix}}={\begin{vmatrix}3&7\\1&{-4}\end{vmatrix}}=3\cdot (-4)-7\cdot 1=-19.$

First properties

The determinant has several key properties that can be proved by direct evaluation of the definition for $2\times 2$-matrices, and that continue to hold for determinants of larger matrices. They are as follows: first, the determinant of the identity matrix ${\begin{pmatrix}1&0\\0&1\end{pmatrix}}$ is 1. Second, the determinant is zero if two rows are the same:

${\begin{vmatrix}a&b\\a&b\end{vmatrix}}=ab-ba=0.$

This holds similarly if the two columns are the same. Moreover,

${\begin{vmatrix}a&b+b'\\c&d+d'\end{vmatrix}}=a(d+d')-(b+b')c={\begin{vmatrix}a&b\\c&d\end{vmatrix}}+{\begin{vmatrix}a&b'\\c&d'\end{vmatrix}}.$

Finally, if any column is multiplied by some number $r$ (i.e., all entries in that column are multiplied by that number), the determinant is also multiplied by that number:

${\begin{vmatrix}r\cdot a&b\\r\cdot c&d\end{vmatrix}}=rad-brc=r(ad-bc)=r\cdot {\begin{vmatrix}a&b\\c&d\end{vmatrix}}.$

Definition

In the sequel, A is a square matrix with n rows and n columns, so that it can be written as

$A={\begin{bmatrix}a_{1,1}&a_{1,2}&\dots &a_{1,n}\\a_{2,1}&a_{2,2}&\dots &a_{2,n}\\\vdots &\vdots &\ddots &\vdots \\a_{n,1}&a_{n,2}&\dots &a_{n,n}\end{bmatrix}}.$

The entries $a_{1,1}$ etc. are, for many purposes, real or complex numbers. As discussed below, the determinant is also defined for matrices whose entries are elements in more abstract algebraic structures known as commutative rings.

The determinant of A is denoted by det(A), or it can be denoted directly in terms of the matrix entries by writing enclosing bars instead of brackets:

${\begin{vmatrix}a_{1,1}&a_{1,2}&\dots &a_{1,n}\\a_{2,1}&a_{2,2}&\dots &a_{2,n}\\\vdots &\vdots &\ddots &\vdots \\a_{n,1}&a_{n,2}&\dots &a_{n,n}\end{vmatrix}}.$

There are various equivalent ways to define the determinant of a square matrix A, i.e. one with the same number of rows and columns: the determinant can be defined via the Leibniz formula, an explicit formula involving sums of products of certain entries of the matrix. The determinant can also be characterized as the unique function depending on the entries of the matrix satisfying certain properties. This approach can also be used to compute determinants by simplifying the matrices in question.

Leibniz formula

The Leibniz formula for the determinant of a 3 × 3 matrix is the following:

${\begin{aligned}{\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}&=a(ei-fh)-b(di-fg)+c(dh-eg)\\&=aei+bfg+cdh-ceg-bdi-afh.\end{aligned}}$

The rule of Sarrus is a mnemonic for this formula: the sum of the products of three diagonal north-west to south-east lines of matrix elements, minus the sum of the products of three diagonal south-west to north-east lines of elements, when the copies of the first two columns of the matrix are written beside it as in the illustration:

This scheme for calculating the determinant of a 3 × 3 matrix does not carry over into higher dimensions.

n × n matrices

The Leibniz formula for the determinant of an $n\times n$-matrix $A$ is a more involved, but related expression. It is an expression involving the notion of permutations and their signature. A permutation of the set $\{1,2,\dots ,n\}$ is a function $\sigma$ that reorders this set of integers. The value in the $i$-th position after the reordering $\sigma$ is denoted by $\sigma _{i}$. The set of all such permutations, the so-called symmetric group, is denoted $S_{n}$. The signature of $\sigma$ is defined to be $+1$ whenever the reordering given by σ can be achieved by successively interchanging two entries an even number of times, and $-1$ whenever it can be achieved by an odd number of such interchanges. Given the matrix $A$ and a permutation $\sigma$, the product

$a_{1,\sigma _{1}}\cdot a_{2,\sigma _{2}}\cdot \dots \cdot a_{n,\sigma _{n}}$

is also written more briefly using Pi notation as

$\prod _{i=1}^{n}a_{i,\sigma _{i}}$.

Using these notions, the definition of the determinant using the Leibniz formula is then

$\det(A)=\sum _{\sigma \in S_{n}}\left(\operatorname {sgn}(\sigma )\prod _{i=1}^{n}a_{i,\sigma _{i}}\right),$

a sum involving all permutations, where each summand is a product of entries of the matrix, multiplied with a sign depending on the permutation.

The following table unwinds these terms in the case $n=3$. In the first column, a permutation is listed according to its values. For example, in the second row, the permutation $\sigma$ satisfies $\sigma _{1}=1,\sigma _{2}=3,\sigma _{3}=2$. It can be obtained from the standard order (1, 2, 3) by a single exchange (exchanging the second and third entry), so that its signature is $\operatorname {sgn}(\sigma )=-1$.

Permutations of $\{1,2,3\}$ and their contribution to the determinant

Permutation $\sigma$	$\operatorname {sgn} (\sigma )$	$\operatorname {sgn}(\sigma )\prod _{i=1}^{n}a_{i,\sigma _{i}}$
1, 2, 3	$+1$	$+a_{1,1}a_{2,2}a_{3,3}$
1, 3, 2	$-1$	$-a_{1,1}a_{2,3}a_{3,2}$
3, 1, 2	$+1$	$+a_{1,3}a_{2,1}a_{3,2}$
3, 2, 1	$-1$	$-a_{1,3}a_{2,2}a_{3,1}$
2, 3, 1	$+1$	$+a_{1,2}a_{2,3}a_{3,1}$
2, 1, 3	$-1$	$-a_{1,2}a_{2,1}a_{3,3}$

The sum of the six terms in the third column then reads

$\sum _{\sigma \in S_{3}}\operatorname {sgn}(\sigma )\prod _{i=1}^{n}a_{i,\sigma _{i}}=a_{1,1}a_{2,2}a_{3,3}-a_{1,1}a_{2,3}a_{3,2}+a_{1,3}a_{2,1}a_{3,2}-a_{1,3}a_{2,2}a_{3,1}+a_{1,2}a_{2,3}a_{3,1}-a_{1,2}a_{2,1}a_{3,3}.$

This gives back the formula for $3\times 3$-matrices above. For a general $n\times n$-matrix, the Leibniz formula involves $n!$ (n factorial) summands, each of which is a product of n entries of the matrix.

The Leibniz formula can also be expressed using a summation in which not only permutations, but all sequences of $n$ indices in the range $1,\dots ,n$ occur. To do this, one uses the Levi-Civita symbol $\varepsilon _{i_{1},\cdots ,i_{n}}$ instead of the sign of a permutation

$\det(A)=\sum _{i_{1},i_{2},\ldots ,i_{n}=1}^{n}\varepsilon _{i_{1}\cdots i_{n}}a_{1,i_{1}}\cdots a_{n,i_{n}},$

This gives back the formula above since the Levi-Civita symbol is zero if the indices $i_{1},\dots ,i_{n}$ do not form a permutation.

Resources

• n x n Determinant, Khan Academy
• 3 x 3 Determinant, Khan Academy
• Determinants Along Other Rows and Columns, Khan Academy
• Rules of Sarrus of Determinant, Khan Academy

References

1. Lang, Serge (1985), Introduction to Linear Algebra, Undergraduate Texts in Mathematics (2 ed.), Springer, ISBN 9780387962054
2. McConnell (1957). Applications of Tensor Analysis. Dover Publications. pp. 10–17.
3. Harris, Frank E. (2014), Mathematics for Physical Science and Engineering, Elsevier, ISBN 9780128010495