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Determinant

The determinant is an operation defined for square matrices. For a square matrix \(A\), \(\det A\) denotes the determinant of \(A\).

This article introduces three definitions of the determinant. It can be proven that the definitions presented here are equivalent.

Definition by Permutations

Prerequisites: Permutation, Inversion Number.

This method is suitable for manually computing determinants of low orders, but its time complexity is factorial.

Let \(\pi(j_1j_2\cdots j_n)\) denote the inversion number of the permutation \(j_1j_2\cdots j_n\), and let \(S_n\) be the set of all permutations of length \(n\). The notation:

\[ \begin{aligned} \det A &= \begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & \vdots & & \vdots\\ a_{n1} & a_{n2} & \cdots & a_{nn}\\ \end{vmatrix} \\ &= \sum_{(j_1j_2\cdots j_n) \in S_n} (-1)^{\pi(j_1j_2\cdots j_n)} a_{1 j_1} a_{2 j_2}\dots a_{n j_n} \end{aligned} \]

represents the \(n\)-th order determinant as an algebraic sum of \(n!\) terms. Each term is the product of \(n\) elements taken from different rows and different columns of the matrix \(A\): \(a_{1j_1}a_{2j_2}\cdots a_{nj_n}\).

The sign in front of the term \(a_{1j_1}a_{2j_2}\cdots a_{nj_n}\) is \({(-1)}^{\pi(j_1j_2\cdots j_n)}\). That is, the sign is positive when \(j_1j_2\cdots j_n\) is an even permutation, and negative when it is an odd permutation.

The diagonal rule for second and third-order determinants is essentially based on this permutation definition. The diagonal rule does not apply to determinants of order four or higher for the same reason. In particular, the determinant of a \(1 \times 1\) matrix is simply the element itself.

Theorem: Consider the product of elements taken from rows \(i_1,i_2,\cdots,i_n\) and columns \(j_1,j_2,\cdots,j_n\) of an \(n\)-th order determinant:

\[ a_{i_1j_1}a_{i_2j_2}\cdots a_{i_nj_n} \]

Here, both \(i_1,i_2,\cdots,i_n\) and \(j_1,j_2,\cdots,j_n\) are permutations of the numbers \(1,2,\cdots,n\). Then the sign of this term in the determinant is \({(-1)}^{s+t}\), where

\[ s=\pi(i_1i_2\cdots i_n) \]
\[ t=\pi(j_1j_2\cdots j_n) \]

Theorem: A determinant is equal to its transpose.

Theorem: Suppose all elements in the \(i\)-th row of the determinant \(\det A\) can be expressed as the sum of two terms:

\[ \begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ \vdots & \vdots & & \vdots\\ b_{i1}+c_{i1} & b_{i2}+c_{i2} & \cdots & b_{in}+c_{in}\\ \vdots & \vdots & & \vdots\\ a_{n1} & a_{n2} & \cdots & a_{nn}\\ \end{vmatrix} \]

Then the determinant equals the sum of two determinants \(\det A_1\) and \(\det A_2\). Here, the \(i\)-th row of \(A_1\) is \(b_{i1},b_{i2},\cdots,b_{in}\), the \(i\)-th row of \(A_2\) is \(c_{i1},c_{i2},\cdots,c_{in}\), and all other rows of \(A_1\) and \(A_2\) are the same as in \(A\). The same property holds for columns.

Inductive Definition

This method only describes an algebraic property of determinants. Its time complexity is also factorial, making it unsuitable for computation.

Cofactor

In an \(n\)-th order determinant \(\det A\), arbitrarily select \(k\) rows and \(k\) columns of matrix \(A\). The \(k \times k\) matrix formed by the elements at the intersection of these rows and columns is called a \(k\)-th order submatrix, and its determinant is called a \(k\)-th order minor.

For an \(n\)-th order determinant \(\det A\), the minor matrix \(M_{ij}\) of an element \(a_{ij}\) is the \((n-1) \times (n-1)\) submatrix obtained by deleting the \(i\)-th row and \(j\)-th column from the original matrix \(A\); its determinant \(\det M_{ij}\) is called the minor.

For an \(n\)-th order determinant \(\det A\), the minor \(\det M_{ij}\) of element \(a_{ij}\) multiplied by the sign \({(-1)}^{i+j}\) is called the cofactor of \(a_{ij}\), denoted by \(A_{ij}\).

From the permutation definition in the previous section, we can deduce:

Theorem: If in an \(n\)-th order determinant \(\det A\), all elements in the \(i\)-th row or the \(j\)-th column except \(a_{ij}\) are \(0\), then the determinant equals the product of \(a_{ij}\) and its cofactor \(A_{ij}\).

Determinant Expansion

Since the determinant is unchanged under matrix transposition, it suffices to describe expansion along either rows or columns.

The determinant \(\det A\) is defined as the sum of the products of all elements in any row (or column) with their corresponding cofactors.

In other words, the determinant can be defined recursively using expansion along a row (or column):

\[ \begin{aligned} \det A &= a_{i1}A_{i1}+a_{i2}A_{i2}+\cdots+a_{in}A_{in} \\ &= \sum_{j = 1}^{n} a_{ij}A_{ij} \\ &= \sum_{j = 1}^{n} (-1)^{i + j} a_{ij} \det M_{ij} \end{aligned} \]
\[ \begin{aligned} \det A &= a_{1j}A_{1j}+a_{2j}A_{2j}+\cdots+a_{nj}A_{nj} \\ &= \sum_{i = 1}^{n} a_{ij}A_{ij} \\ &= \sum_{i = 1}^{n} (-1)^{i + j} a_{ij} \det M_{ij} \end{aligned} \]

The base case of the recursion is the determinant of a \(1 \times 1\) matrix, which is simply the single element it contains.

From this, we have:

Theorem: The sum of the products of the elements of one row (or column) of a determinant \(\det A\) with the cofactors of the corresponding elements of another row (or column) equals \(0\).

In other words, when \(i\neq j\):

\[ a_{i1}A_{j1}+a_{i2}A_{j2}+\cdots+a_{in}A_{jn}=0 \]
\[ a_{1i}A_{1j}+a_{2i}A_{2j}+\cdots+a_{ni}A_{nj}=0 \]

Axiomatic Definition

The axiomatic definition states that any operation satisfying certain properties must be the determinant.

Prerequisite: Elementary Operations.

Let \(D_i(k)\) denote the scaling matrix, \(P_{ij}\) denote the swap matrix, and \(T_{ij}(k)\) denote the row addition matrix.

For an operation \(\det\) on an \(n \times n\) matrix \(A\), if it satisfies the following four properties, it is called the determinant:

  • Multiplying all elements of one row or one column of a determinant by a number \(k\) is equivalent to multiplying the determinant by \(k\).

    \[ \det(D_i(k)A) = \det(AD_i(k)) = k \det A \]

  • Swapping two rows or two columns of a determinant changes its sign.

    \[ \det(P_{ij}A) = \det(AP_{ij}) = -\det A \]

  • Adding a multiple of one row (or column) to another row (or column) leaves the determinant unchanged.

    \[ \det(T_{ij}(k)A) = \det(AT_{ij}(k))= \det A \]

  • The determinant of the identity matrix is \(1\).

    \[ \det I = 1 \]

Using the properties of determinants related to elementary operations, it is convenient to manually compute determinants of higher orders. The Gaussian elimination method for computing determinants also utilizes these properties, with a time complexity of \(O(n^3)\).

Several corollaries follow from these properties:

  • A common factor of all elements in a row or column can be factored out of the determinant.
  • If all elements in a row or column of a determinant are \(0\), then the determinant is \(0\).
  • If two rows or two columns of a determinant are proportional, then the determinant is \(0\).
  • If two rows or two columns of a determinant are identical, then the determinant is \(0\).

These corollaries are very useful when computing determinants by hand.

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