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Lecture 2: Definition and properties of determinants

Postgraduate Entrance Exam Mathematics Study Notes: Lecture 2: Definition and Properties of Determinants. Retain original formulas, diagrams, and example problems.

Mathematics18 min read
Graduate Entrance Examination MathematicsLinear AlgebraDeterminants
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2.1 Basic Concepts of Determinants

2.1.1 Origin of Determinants

Origin of Determinants

  • Concept:
  • Matrix:
  • $$\left(\begin{matrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{matrix}\right)=A$$
  • Determinant:
  • $$\begin{vmatrix}\mathrm{a}&\mathrm{b}\\\mathrm{c}&\mathrm{d}\end{vmatrix}=|A|$$
  • Relationships:
  • The concept of determinants comes from matrices, which are apropertyof transformations;
  • A matrix is used toexpress information; only when acting on one vector will it act on another;
  • A determinant is a measure of a certain property or feature by placing matrix A in;

Supplement: The meaning of a determinant in two dimensions

  • Concept:
  • The determinant gives the signed area-scaling factor of the linear transformation represented by a matrix;
  • Example:
  • For a two-dimensional matrix $A$, $\det(A)=0$ means that the transformation compresses the plane into a line or a point;
  • Consequently, any planar figure transformed by $A$ has area 0;
  • Meaning:
  • Thus, checking whether the determinant is 0 tells us whether the transformation collapses the space into a lower dimension; equivalently, the matrix columns are linearly dependent;
  • Supplement: Orientation
  • A negative determinant reverses orientation;
  • Supplement: Three dimensions
  • In three dimensions, the absolute value of the determinant is the volume-scaling factor of the linear transformation;
  • Therefore, if the determinant of a $3\times3$ matrix is 0, its columns are linearly dependent;

Concept: Calculation of second-order determinants

  • Formula:
  • $$|A|=|\begin{matrix}a&b\\c&d\end{matrix}|=ad-bc$$
  • Main Diagonal:
  • From a->d is the main diagonal;
  • Subdiagonal:
  • From b->c is the secondary diagonal;
  • The value of the determinant:
  • It is the product of the main-diagonal entries minus the product of the secondary-diagonal entries;
  • Geometrically, its absolute value measures the area-scaling factor of the corresponding linear transformation;

2.1.2 Determinant Knowledge Structure

Knowledge Structure

  • Basic concepts of determinants:
  • Essential definition
  • Properties
  • Definition using inversion counts
  • Expansion theorem
  • Several important determinants: 12+1
  • Main diagonal determinant
  • Determinant of the subdiagonal
  • Laplace expansion
  • Vandermonde determinant
  • Calculation of determinants
  • Concrete type
  • Transform into the basic type
  • Recurrence method
  • Functions and equations represented by determinants
  • Abstract type
  • Apply determinant properties
  • $|AB|=|A||B|$ using the formula
  • Supplement: AB does not equal BA
  • But |AB| result of the area equals the areas of the determinants A and B, and it is worth multiplying
  • So: |AB|=|A||B|=|B||A|=|BA|
  • Calculation of cotons and algebraic coders
  • Kramer's Law
  • Not important

2.2 The first definition of determinants

2.2.1 Second-Order Determinant

Concept: The concept of second-order determinants

  • Concept:
  • $$2 \text{ The steps and rows are arranged in order }D_2=\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix}$$
  • Explanation:
  • The first subscript of $a_{ij}$ $i$ indicates the number of rows where the element is located, and the second indication $j$ indicates the number of columns -> the fixed position;
  • $i=1,2,j=1,2$；
  • Thus, there are four elements in this determinant;
  • Calculation:
  • $\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix}=a_{11}a_{22}-a_{12}a_{21}$
  • Geometric meaning:
  • Calculation:
  • $S_D=lm\sin(\beta-\alpha)=a_{11}a_{22}-a_{12}a_{21}$
  • Meaning:
  • The second-order determinant is the area value of a parallelogram -> reflects the measure;
  • 2 A determinant consists oftwo 2-dimensional vectors, and its (according to the operation rules) result is the area of a parallelogram with these two vectors as adjacent sides. This not only derives the calculation rules for the second-order determinant, but also clearly shows its geometric meaning;

2.2.2 Third-Order Determinant

Concept: Third-order determinant

  • Concept:
  • $$\text{ 3rd order determinant }D_{3}=\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}$$
  • Explanation:
  • 3-order determinant consists of three 3-dimensional vectors $a_1=[a_{11}, a_{12}, a_{13}]$, $a_2=[a_{21}, a_{22}, a_{23}]$, and $a_3=[a_{31}, a_{32}, a_{33}]$;
  • Its (operation rules) result isthe volume of a parallelepiped with these three vectors as adjacent sides;
  • Calculation:
  • Formula $$\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}=(a_{11}a_{22}a_{33}+a_{13}a_{21}a_{32}+a_{12}a_{23}a_{31})-(a_{13}a_{22}a_{31}+a_{12}a_{21}a_{33}+a_{11}a_{23}a_{32})$$
  • Manual Calculation Method:
  • Line drawing method -> draw two lines. The first line starts from a11->a22->a33, then based on this line, draw two forks: a13->a21->a32 and a12->a23->a31; Similarly, the second line uses a13->a22->a31 and then draws two crosses;

2.2.3 Essential Definition of an N-Order Determinant

#####Definition: nThe steps and rows are arranged in order

description: $$\begin{aligned}&n\text{ The steps and rows are arranged in order }D_n=\begin{vmatrix}a_{11}&\cdots&a_{1n}\\\vdots&&\vdots\\a_{n1}&\cdots&a_{nn}\end{vmatrix}\text{ It was because of this }n\text{ Piece }n\text{ Dimension vector }a_1=\begin{bmatrix}a_{11}&a_{12}&\cdots,a_{1n}\end{bmatrix},\cdots,a_{n}=[a_{n1},a_{n2},\cdots,a_{nn}]\text{ , and its (operation rule) result is given by this }n\text{ The vectors are adjacent sides }n\text{ Dimensional graphical (yes }\\&\text{ Volume. }\end{aligned}$$

Explanation

  • The determinant is 0, the set of vectors is linearly dependent, the determinant is not 0, the set of vectors is linearly independent;

2.3 The second definition of the determinant

2.3.1 Permutations and Inversions

Concept: Permutation

  • An ordered arrangement of the numbers 1,2,…,n is called a permutation of order $n$.
  • For example, 23145 and 41352 are permutations of order $5$. There are $n!$ permutations of order $n$.

Concept: Inversion

  • In a permutation $i_1i_2\cdots i_s\cdots i_t\cdots i_n$, if $s<t$ but $i_s>i_t$, then the pair $(i_s,i_t)$ is called an inversion.

Concept: Inversion number

  • The total number of inversions in a permutation is its inversion number, denoted by $\tau\left(i_{1}i_{2}\cdots i_{n}\right)$. For example, $\tau(231546)=3$.
  • The natural order 12345 has inversion number 0.

Concept: Odd-permutation and even-permutation

  • If the reverse order of the permutation is odd, it is called an odd permutation;
  • When the reverse order of the permutation is even, it is called an even permutation;

2.3.2 The second definition of an n-th-order determinant

#####Definition: nThe steps and rows are arranged in order: The second definition

description: $$\begin{vmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\a_{21}&a_{22}&\cdots&a_{2n}\\\vdots&\vdots&&\vdots\\a_{n1}&a_{n2}&\cdots&a_{nn}\end{vmatrix}=\sum_{j_{1}j_{2}\cdots j_{n}}(-1)^{r(j_{1}j_{2}\cdots j_{n})}a_{1j_{1}}a_{2j_{2}}\cdots a_{nj_{n}}.$$

Explanation

  • Explanation:
  • $\sum_{h_1,h_2,...h_n}$：
  • Represents the sum of all $n$ columns arranged by index, thus summing $n!$ terms;
  • Note that -> row indexes are arranged in order, while column indexes are arranged in any $n$ level;
  • $(-1)^{\tau(j_1j_2\cdots j_n)}.$：
  • The positive and negative signs for each item depend on $(-1)^{\tau(j_1j_2\cdots j_n)}.$
  • That is: the positive or negative depends on the result of $\tau(j_1j_2\cdots j_n)$, and the reverse order of the column's indices is calculated by permutation;
  • When column indices are arranged odd, a negative sign should be appended; When column indices are even, a positive sign should be appended;
  • $a_{1j_{1}}a_{2j_{2}}\cdots a_{nj_{n}}$：
  • Each term consists of the product of $n$ elements taken from different rows and columns
  • Meaning:
  • Expand the n-order determinant into n factorial term -> represent the nth-order permutation, a general calculation rule;
  • Example:
  • Title:
  • $\text{ Please confirm }a_{12}a_{31}a_{54}a_{43}a_{25}\text{ The positive and negative signs before this expansion item }$
  • Analysis:
  • First, arrange the rows -> $a_{12}a_{25}a_{31}a_{43}a_{54}$
  • Then look at the $\tau(25134)=4$

2.4 The third definition of determinants

Core Idea: Downgrade

2.4.1 Minors

#####Definition: Minor

Description: In a determinant of order $n$, delete row $i$ and column $j$ containing $a_{ij}$. The determinant of order $n-1$ formed by the remaining elements is the minor of $a_{ij}$, denoted by $M_{ij}$.
That is: $$M_{ij}=\begin{vmatrix}a_{11}&\cdots&a_{1,j-1}&a_{1,j+1}&\cdots&a_{1n}\\\vdots&&\vdots&\vdots&&\vdots\\a_{i-1,1}&\cdots&a_{i-1,j-1}&a_{i-1,j+1}&\cdots&a_{i-1,n}\\a_{i+1,1}&\cdots&a_{i+1,j-1}&a_{i+1,j+1}&\cdots&a_{i+1,n}\\\vdots&&\vdots&\vdots&&\vdots\\a_{n1}&\cdots&a_{n,j-1}&a_{n,j+1}&\cdots&a_{nn}\end{vmatrix}.$$

Explanation

  • Supplement:
  • Minor:
  • A minor is itself a determinant formed from selected rows and columns of the original matrix or determinant.
  • Submatrix:
  • A submatrix is a matrix;
  • Explanation:
  • Deleting row $i$ and column $j$ leaves an $(n-1)\times(n-1)$ determinant, namely the minor $M_{ij}$.
  • Example:
  • For $\begin{vmatrix}1&2&1\\2&5&7\\-6&4&9\end{vmatrix}$, the minor of the entry in row $2$, column $3$ is $M_{23}=\left|\begin{matrix}1&2\\-6&4\end{matrix}\right|$.

2.4.2 Cofactors

#####Definition: Cofactor

Description: Multiplying the minor $M_{ij}$ by $(-1)^{i+j}$ gives the cofactor of $a_{ij}$, denoted by $A_{ij}$:
$$A_{ij}=(-1)^{i+j}M_{ij},\qquad M_{ij}=(-1)^{i+j}A_{ij}.$$

Explanation

  • Concept:
  • A cofactor is a minor with the sign factor $(-1)^{i+j}$.
  • Example:
  • $A_{23}=(-1)^{2+3}M_{23}$

2.4.3 Determinant Expansion Formula

#####Theorem: Theexpansiontheoremofdeterminants

Description: the value of a determinant equals the value of a certain row (column) element of the determinant, multiply it by its corresponding algebraic coefficient expression, then sum, i.e., $$\left|A\right|=\begin{cases}a_{i1}A_{i1}+a_{i2}A_{i2}+\cdots+a_{in}A_{in}=\sum_{j=1}^{n}a_{ij}A_{ij}\left(i=1,2,\cdots,n\right),\\\\a_{1j}A_{1j}+a_{2j}A_{2j}+\cdots+a_{nj}A_{nj}=\sum_{i=1}^{n}a_{ij}A_{ij}\left(j=1,2,\cdots,n\right).\end{cases}$$

Explanation

  • Example:
  • Expand:
  • For $|A|_{3\times3}$, expand according to the first line to get $|A|_{3\times3}=a_{11}A_{11}+a_{12}A_{12}+a_{13}A_{13}$
  • Analysis:
  • In the example above, a third-order expression is expanded into second-order, but the trade-off is that the expression becomes three;
  • If at fourth order it is $|A|_{4\times4}=a_{21}A_{21}+a_{22}A_{22}+a_{23}A_{3}+a_{24}A_{24}$
  • Purpose:
  • Core Idea -> Find the value of the determinant after descending the order;
  • Downgrade n steps into n n-1 steps
  • Principles of Development:
  • The more elements in a row (column) of 0, the better;

2.5 Properties of Determinants

2.5.1 Property One

Property One: Row and column swaps, with the value unchanged, i.e., $|A|=|A^{\mathrm{T}}|$

  • Concept Supplement:
  • Transpose Matrix:
  • If $|A|=|A^{\mathrm{T}}|$, then $A^{\mathrm{T}}$ is called thetranspose matrixof A;
  • Symmetric Matrices:
  • On the basis of $A^{\mathrm{T}}$ is atranspose matrixof A, if $A=A^{\mathrm{T}}$, then it is called a symmetric matrix;
  • Nature Explanation:
  • $|A|=|A^{\mathrm{T}}|$ -> Even if the current $A^{\mathrm{T}}$ and A are not symmetric matrices, their values (the geometric area of the second-order determinant) are the same;
  • Conclusion:
  • For determinants, the status of rows and columns is equivalent ->can be used for properties on rows, and both can be used withon columns;

2.5.2 Property Two

Property Two: If all elements in a determinant are zero, then the determinant is zero

  • If in the determinant there is a row 0 -> means the determinant has been reduced by one dimension -> Because it cannot be measured, the measure is 0;
  • Similarly, if a column is all zero, then the determinant is zero;

2.5.3 Property Three: Multiplication

Property three: If a row (column) element in determinant has a common factor k(k≠0), then k can be factored out of the determinant, that is:

  • Formula:
  • $$\begin{vmatrix}a_{11}&a_{12}&\ldots&a_{1n}\\\vdots&\vdots&&\vdots\\ka_{i1}&ka_{i2}&\ldots&ka_{in}\\\vdots&\vdots&&\vdots\\a_{n1}&a_{n2}&\ldots&a_{nn}\end{vmatrix}=k\begin{vmatrix}a_{11}&a_{12}&\ldots&a_{1n}\\\vdots&\vdots&&\vdots\\a_{i1}&a_{i2}&\ldots&a_{in}\\\vdots&\vdots&&\vdots\\a_{n1}&a_{n2}&\ldots&a_{nn}\end{vmatrix}.$$
  • Explanation:
  • If understood geometrically: if a second-order determinant is multiplied by K, it means it multiplies by k in a certain direction, not all (here, all of which are actually two directions), and can only extend one side -> If multiplied by K for all directions, the result is squared;
  • In the process of extracting K by multiplying above, operations from right to left of the equation are called "multiplication" properties;
  • Example:
  • You can multiply (extract) the number onto a certain line
  • $$\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix}=k\begin{vmatrix}a_{11}&a_{12}\\\frac{a_{21}}{k}&\frac{a_{22}}{k}\end{vmatrix}$$
  • Function:
  • Remove the denominator from the determinant;

2.5.4 Property Four

Property 4: If a certain row (column) element in a determinant is the sum of two elements, it can be split into the sum of two determinants, that is:

  • Formula:
  • $$\begin{vmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\\vdots&\vdots&&\vdots\\a_{i1}+b_{i1}&a_{i2}+b_{i2}&\cdots&a_{in}+b_{in}\\\vdots&\vdots&&\vdots\\a_{n1}&a_{n2}&\cdots&a_{nn}\end{vmatrix}=\begin{vmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\\vdots&\vdots&&\vdots\\a_{i1}&a_{i2}&\cdots&a_{m}\\\vdots&\vdots&&\vdots\\a_{n1}&a_{n2}&\cdots&a_{nn}\end{vmatrix}+\begin{vmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\\vdots&\vdots&&\vdots\\b_{i1}&b_{i2}&\cdots&b_{in}\\\vdots&\vdots&&\vdots\\a_{n1}&a_{n2}&\cdots&a_{nn}\end{vmatrix}$$
  • Explanation:
  • Single-row detachable (add-on) capability;
  • The equation from right to left is the sum of two determinants. Ifthe other elements of two determinants correspond to equal and only one row (column) is different, you can add them. When adding, the other elements remain unchanged, and the rows (columns) of different elements are added accordingly;
  • Supplement:
  • Additivity for single lines -> Different rows in the determinant, but every other line is exactly the same -> Only then can additions be made;
  • Example: When using a second-order determinant
  • $$\begin{vmatrix}a_1&a_2\\b_1&b_2\end{vmatrix}+\begin{vmatrix}a_1&a_2\\c_1&c_2\end{vmatrix}=\begin{vmatrix}a_1&a_2\\c_1+b_1&c_2+b_2\end{vmatrix}$$
  • Note:
  • Obviously, by this property we know: $|A|+|B|\neq|A+B|$

2.5.5 Property Five: Interchange

Property Five: In a determinant, two rows (or columns) are swapped, and the determinant is changed

  • Geometric meaning:
  • For example, in a second-order determinant, if the parallelogram drawn and the sides are swapped in order, the resulting area is thus reversed;
  • Note:
  • This property is called the "swap" property;
  • When swapping, if the value of -> remains unchanged after even swaps; If the value changes -> odd times;

2.5.6 Property Six

Property 6: If two rows (columns) in a determinant are equal or proportional, then the determinant is zero

  • Example:
  • $$\vert\begin{matrix}1&2\\2&4\end{matrix}\vert=0$$
  • Explanation:
  • If two rows of elements are exactly the same or proportional, then the determinant is zero;
  • Geometric Meaning:
  • In a second-order determinant, the two vectors are parallel;
  • Note:
  • The negative direction can also be parallel;

2.5.7 Property Seven: Multiplication

Property 7: In a determinant, the k multiplied by one row (column) to another row (column) remains unchanged

  • Example:
  • Multiply the -2 of the $|1\quad 2|$ in the $\left|\begin{matrix}1&2\\2&3\end{matrix}\right|$ into the $|2\quad 3|$
  • That is: (1,2)*-2=(-2,-4) is added to $|2\quad 3|$ -> |0，-1| -> to get $\left|\begin{matrix}1&2\\0&-1\end{matrix}\right|$
  • and $\left|\begin{matrix}1&2\\0&-1\end{matrix}\right|$ = $\left|\begin{matrix}1&2\\2&3\end{matrix}\right|$
  • Explanation:
  • This property is called the "multiplication" property -> It involves multiplying one line and then adding it to another line;
  • Function:
  • Transform the 0 in the determinant into -> to simplify it;

2.6 Several Important Determinants

2.6.1 Main Diagonal Determinant

#####Definition: Maindiagonaldeterminant

description: $$\begin{vmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\0&a_{22}&\cdots&a_{2n}\\\vdots&\vdots&&\vdots\\0&0&\cdots&a_{nn}\end{vmatrix}=\begin{vmatrix}a_{11}&0&\cdots&0\\a_{21}&a_{22}&\cdots&0\\\vdots&\vdots&&\vdots\\a_{n1}&a_{n2}&\cdots&a_{nn}\end{vmatrix}=\begin{vmatrix}a_{11}&0&\cdots&0\\0&a_{22}&\cdots&0\\\vdots&\vdots&&\vdots\\0&0&\cdots&a_{nn}\end{vmatrix}=\prod_{i=1}^na_{ii}.$$

Explanation: The determinant of the upper (lower) triangle

  • The result equals the product of the main diagonal elements;

2.6.2 Determinant of the Paradiagonal

#####Definition: Thedeterminantofthesubdiagonal

description: $$\begin{aligned}\begin{vmatrix}a_{11}&a_{12}&\cdots&a_{1,n-1}&a_{1n}\\a_{21}&a_{22}&\cdots&a_{2,n-1}&0\\\vdots&\vdots&&\vdots&\vdots\\a_{n1}&0&\cdots&0&0\end{vmatrix}&=\begin{vmatrix}0&\cdots&0&a_{1n}\\0&\cdots&a_{2,n-1}&a_{2n}\\\vdots&&\vdots&\vdots\\a_{n1}&\cdots&a_{n,n-1}&a_{nn}\end{vmatrix}=\begin{vmatrix}0&\cdots&0&a_{1n}\\0&\cdots&a_{2,n-1}&0\\\vdots&&\vdots&\vdots\\a_{n1}&\cdots&0&0\end{vmatrix}\\&=(-1)^{\frac{n(n-1)}{2}}a_{1n}a_{2,n-1}\cdots a_{n1}.\end{aligned}$$

Explanation

  • The result equals the subdiagonal elements multiplied by the inverse ordinal;

2.6.3 Laplace Expansion

#####Definition: LaplaceExpandableStyle

Description: $$\begin{gathered}

Suppose A is an m-order matrix and B is an n-order matrix, then \\

\begin{vmatrix}A&O\\O&B\end{vmatrix}=\begin{vmatrix}A&C\\O&B\end{vmatrix}=\begin{vmatrix}A&O\\C&B\end{vmatrix}=|A||B|, \\

\\

\begin{vmatrix}\boldsymbol{O}&\boldsymbol{A}\\\boldsymbol{B}&\boldsymbol{O}\end{vmatrix}=\begin{vmatrix}\boldsymbol{C}&\boldsymbol{A}\\\boldsymbol{B}&\boldsymbol{O}\end{vmatrix}=\begin{vmatrix}\boldsymbol{O}&\boldsymbol{A}\\\boldsymbol{B}&\boldsymbol{C}\end{vmatrix}=(-1)^{mn}|\boldsymbol{A}||\boldsymbol{B}|.

\end{gathered}$$

Explanation

  • Determinant of a block matrix: $\begin{vmatrix}A&O\\O&B\end{vmatrix}$
  • All A、B、0 here are matrices;
  • Using several horizontal lines and several vertical lines, divide a matrix into several submatrices. The resulting sub-matrices are chunks of the original matrix. We call this method of matrix representation called a chunking matrix;
  • For example: A can be a third-order matrix $A_{33}$, B can be a second-order matrix $B_{22}$, so $\begin{vmatrix}A&O\\O&B\end{vmatrix}$ is a fifth-order matrix;
  • Meaning of block matrix:
  • Generalize the matrix from a numerical perspective;
  • Analysis:
  • $\begin{vmatrix}A&O\\O&B\end{vmatrix}=\begin{vmatrix}A&C\\O&B\end{vmatrix}=\begin{vmatrix}A&O\\C&B\end{vmatrix}=|A||B|$ <- Its conclusion can be understood as the main diagonal conclusion;
  • $\begin{vmatrix}\boldsymbol{O}&\boldsymbol{A}\\\boldsymbol{B}&\boldsymbol{O}\end{vmatrix}=\begin{vmatrix}\boldsymbol{C}&\boldsymbol{A}\\\boldsymbol{B}&\boldsymbol{O}\end{vmatrix}=\begin{vmatrix}\boldsymbol{O}&\boldsymbol{A}\\\boldsymbol{B}&\boldsymbol{C}\end{vmatrix}=(-1)^{mn}|\boldsymbol{A}||\boldsymbol{B}|$ <- Its conclusion can be understood as a conclusion in the form of a subdiagonal, mn is how many times it has changed;

2.6.4 Vandermonde Determinant

Definition: VandermondeDeterminant
Description: $$\begin{vmatrix}1&1&\cdots&1\\x_1&x_2&\cdots&x_n\\x_1^2&x_2^2&\cdots&x_n^2\\\vdots&\vdots&&\vdots\\x_1^{n-1}&x_2^{n-1}&\cdots&x_n^{n-1}\end{vmatrix}=\prod_{1\leq i<j\leq n}(x_j-x_i),n\geq2.$$

Explanation

  • Analysis: A third-order Vandermonde determinant is $V_{3}=\begin{vmatrix}1&1&1\\x_{1}&x_{2}&x_{3}\\x_{1}^{2}&x_{2}^{2}&x_{3}^{2}\end{vmatrix}$;
  • Calculation for the third-order case:
  • Form all differences $x_j-x_i$ with $j>i$;
  • $V_3=(x_{2}-x_{1})(x_{3}-x_{1})(x_{3}-x_{2})$.
  • Note:
  • The last row has a frequency minus one, because the number starts from zero;
  • Example:
  • $\begin{vmatrix}1&a&a^{2}\\1&b&b^{2}\\1&c&c^{2}\end{vmatrix}=\begin{vmatrix}1&1&1\\a&b&c^{2}\\a^{2}&b^{2}&c^{2}\end{vmatrix}=$