hcju/proofwikips huggingface dataset:
Properties of Matrix Entrywise Addition over Ring
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $S$ over an algebraic structure $\struct {R, +, \circ}$. Let $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$. Let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$. The operation of matrix entrywise addition satisfies the following properties: :$+$ is closed on $\map {\MM_R} {m, n}$ :$+$ is associative on $\map {\MM_R} {m, n}$ :$+$ is commutative on $\map {\MM_R} {m, n}$.
Matrix Entrywise Addition is Associative
Let $\map \MM {m, n}$ be a $m \times n$ matrix space over one of the standard number systems. For $\mathbf A, \mathbf B \in \map \MM {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$. The operation $+$ is associative on $\map \MM {m, n}$. That is: :$\paren {\mathbf A + \mathbf B} + \mathbf C = \mathbf A + \paren {\mathbf B + \mathbf C}$ for all $\mathbf A$, $\mathbf B$ and $\mathbf C$ in $\map \MM {m, n}$.
Matrix Entrywise Addition over Ring is Commutative
Let $\struct {R, +, \circ}$ be a ring. Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $R$. For $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$. The operation $+$ is commutative on $\map {\MM_R} {m, n}$. That is: :$\mathbf A + \mathbf B = \mathbf B + \mathbf A$ for all $\mathbf A$ and $\mathbf B$ in $\map {\MM_R} {m, n}$.
Zero Matrix is Identity for Matrix Entrywise Addition
Let $\map \MM {m, n}$ be a $m \times n$ matrix space over one of the standard number systems. Let $\mathbf 0 = \sqbrk 0_{m n}$ be the zero matrix of $\map \MM {m, n}$. Then $\mathbf 0$ is the identity element for matrix entrywise addition.
Matrix Scalar Product is Associative
Let $\Bbb F$ denote one of the standard number systems. Let $\map \MM {m, n}$ be a $m \times n$ matrix space over $\Bbb F$. For $\mathbf A \in \map \MM {m, n}$ and $\lambda$ \in $\Bbb F$, let $\lambda \mathbf A$ be defined as the matrix scalar product of $\lambda$ and $\mathbf A$. The matrix scalar product is associative on $\map \MM {m, n}$, in the following sense: For all $\mathbf A$ in $\map \MM {m, n}$ and $\lambda, \mu \in \Bbb F$: :$\lambda \paren {\mu \mathbf A} = \paren {\lambda \mu} \mathbf A$
Matrix Scalar Product Distributes over Number Addition
Let $\Bbb F$ denote one of the standard number systems. Let $\map \MM {m, n}$ be a $m \times n$ matrix space over $\Bbb F$. For $\mathbf A \in \map \MM {m, n}$ and $\lambda$ \in $\Bbb F$, let $\lambda \mathbf A$ be defined as the matrix scalar product of $\lambda$ and $\mathbf A$. The matrix scalar product is associative on $\map \MM {m, n}$, in the following sense: For all $\mathbf A$ in $\map \MM {m, n}$ and $\lambda, \mu \in \Bbb F$: :$\paren {\lambda + \mu} \mathbf A = \lambda \mathbf A + \mu \mathbf A$
Negative Matrix is Inverse for Matrix Entrywise Addition
Let $\Bbb F$ denote one of the standard number systems. Let $\map \MM {m, n}$ be a $m \times n$ matrix space over $\Bbb F$. Let $\mathbf A$ be an element of $\map \MM {m, n}$. Let $-\mathbf A$ be the negative of $\mathbf A$. Then $-\mathbf A$ is the inverse for the operation $+$, where $+$ is matrix entrywise addition.
Zero Matrix is Zero for Matrix Multiplication
Let $\struct {R, +, \times}$ be a ring. Let $\mathbf A$ be a matrix over $R$ of order $m \times n$ Let $\mathbf 0$ be a zero matrix whose order is such that either: :$\mathbf 0 \mathbf A$ is defined or: :$\mathbf A \mathbf 0$ is defined or both. Then: :$\mathbf 0 \mathbf A = \mathbf 0$ or: :$\mathbf A \mathbf 0 = \mathbf 0$ whenever they are defined. The order of $\mathbf 0$ will be according to the orders of the factor matrices.
Unit Matrix is Identity for Matrix Multiplication
Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\map {\MM_R} n$ denote the metric space of square matrices of order $n$ over $R$. Let $\mathbf I_n$ denote the unit matrix of order $n$: Then: :$\forall \mathbf A \in \map {\MM_R} n: \mathbf A \mathbf I_n = \mathbf A = \mathbf I_n \mathbf A$ That is, the unit matrix $\mathbf I_n$ is the identity element for (conventional) matrix multiplication over $\map {\MM_R} n$.
Unit Matrix is Identity for Matrix Multiplication/Left
Let $\map {\MM_R} {m, n}$ denote the $m \times n$ metric space over $R$. Let $I_m$ denote the unit matrix of order $m$. Then: :$\forall \mathbf A \in \map {\MM_R} {m, n}: \mathbf I_m \mathbf A  = \mathbf A$
Unit Matrix is Identity for Matrix Multiplication/Right
Let $\map {\MM_R} {m, n}$ denote the $m \times n$ metric space over $R$. Let $I_n$ denote the unit matrix of order $n$. Then: :$\forall \mathbf A \in \map {\MM_R} {m, n}: \mathbf A \mathbf I_n  = \mathbf A$
Inverse of Square Matrix over Field is Unique
Let $\Bbb F$ be a field, usually one of the standard number fields $\Q$, $\R$ or $\C$. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\map \MM n$ denote the matrix space of order $n$ square matrices over $\Bbb F$. Let $\mathbf B$ be an inverse matrix of $\mathbf A$. Then $\mathbf B$ is the only inverse matrix of $\mathbf A$.
Rank of Empty Set is Zero
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $\rho : \powerset S \to \Z$ be the rank function of $M$. Then: :$\map \rho \O = 0$
Rank Function is Increasing
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $\rho: \powerset S \to \Z$ be the rank function of $M$. Let $A, B \subseteq S$ be subsets of $S$ such that $A \subseteq B$. Then: :$\map \rho A \le \map \rho B$
Inverse of Transpose of Matrix is Transpose of Inverse
Let $\mathbf A$ be a matrix over a field. Let $\mathbf A^\intercal$ denote the transpose of $\mathbf A$. Let $\mathbf A$ be an invertible matrix. Then $\mathbf A^\intercal$ is also invertible and: :$\paren {\mathbf A^\intercal}^{-1} = \paren {\mathbf A^{-1} }^\intercal$ where $\mathbf A^{-1}$ denotes the inverse of $\mathbf A$.
Max yields Supremum of Parameters/General Case
Let $x_1, x_2, \dots ,x_n \in S$ for some $n \in \N_{>0}$. Then: :$\max \set {x_1, x_2, \dotsc, x_n} = \sup \set {x_1, x_2, \dotsc, x_n}$
Max of Subfamily of Operands Less or Equal to Max
Let $\struct {S, \preceq}$ be a totally ordered set. Let $x_1, x_2, \dotsc, x_n \in S$ for some $n \in \N_{>0}$. Let $\set{k_1, k_2, \dotsc, k_m} \subseteq \set{1, 2, \dotsc, n}$ Then: :$\max \set {x_{k_1}, x_{k_2}, \dotsc, x_{k_m}} \preceq \max \set {x_1, x_2, \dotsc, x_n}$ where: :$\max$ denotes the max operation
Elementary Matrix corresponding to Elementary Row Operation
Let $\mathbf I$ denote the unit matrix of order $m$ over a field $K$. Let $e$ be an elementary row operation on $\mathbf I$. Let $\mathbf E$ be the elementary row matrix of order $m$ uniquely defined as: :$\mathbf E = e \paren {\mathbf I}$ where $\mathbf I$ is the unit matrix. Let $r_k$ denote the $k$th row of $\mathbf I$ for $1 \le k \le m$.
Elementary Matrix corresponding to Elementary Row Operation/Scale Row
Let $e$ be the elementary row operation acting on $\mathbf I$ as: {{begin-axiom}} {{axiom | n = \text {ERO} 1         | t = For some $\lambda \in K_{\ne 0}$, multiply row $k$ of $\mathbf I$ by $\lambda$         | m = r_k \to \lambda r_k }} {{end-axiom}}
Elementary Matrix corresponding to Elementary Row Operation/Scale Row and Add
Let $e$ be the elementary row operation acting on $\mathbf I$ as: {{begin-axiom}} {{axiom | n = \text {ERO} 2         | t = For some $\lambda \in K$, add $\lambda$ times row $j$ to row $i$         | m = r_i \to r_i + \lambda r_j }} {{end-axiom}}
Elementary Matrix corresponding to Elementary Row Operation/Exchange Rows
Let $e$ be the elementary row operation acting on $\mathbf I$ as: {{begin-axiom}} {{axiom | n = \text {ERO} 3         | t = Interchange rows $i$ and $j$         | m = r_i \leftrightarrow r_j }} {{end-axiom}}
Power Set of Singleton
Let $x$ be an object. Then the power set of the singleton $\set x$ is: :$\powerset {\set x} = \set {\O, \set x}$
Row Operation to Clear First Column of Matrix
Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over a field $K$. Then there exists a row operation to convert $\mathbf A$ into another $m \times n$ matrix $\mathbf B = \sqbrk b_{m n}$ with the following properties: :$(1): \quad$ Except possibly for element $b_{1 1}$, all the elements of column $1$ are $0$ :$(2): \quad$ If $b_{1 1} \ne 0$, then $b_{1 1} = 1$. This process is referred to as '''clearing the first column'''.
Matrix is Row Equivalent to Echelon Matrix
Let $\mathbf A = \sqbrk a_{m n}$ be a matrix of order $m \times n$ over a field $F$. Then $A$ is row equivalent to an echelon matrix of order $m \times n$.
Singleton is Independent implies Rank is One
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $x \in S$. Let $\set x$ be independent. Then: :$\map \rho {\set x} = 1$ where $\rho$ denotes the rank function of $M$.
Singleton is Dependent implies Rank is Zero
:$\map \rho {\set x} = 0$
System of Simultaneous Equations may have No Solution
Let $S$ be a system of simultaneous equations. Then it is possible that $S$ may have a solution set which is empty.
System of Simultaneous Equations may have Unique Solution
Let $S$ be a system of simultaneous equations. Then it is possible that $S$ may have a solution set which is a singleton.
Absolutely Convergent Series is Convergent iff Normed Vector Space is Banach
Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be an absolutely convergent series in $X$. Then $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ is convergent  $X$ is a Banach space.
Absolutely Convergent Series is Convergent iff Normed Vector Space is Banach/Necessary Condition
Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be an absolutely convergent series in $X$. Suppose $X$ is a Banach space. Then $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ is convergent.
Absolutely Convergent Series is Convergent iff Normed Vector Space is Banach/Sufficient Condition
Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be an absolutely convergent series in $X$. Suppose $\ds \sum_{n \mathop = 1}^\infty a_n$ is convergent. Then $X$ is a Banach space.
Trivial Solution to System of Homogeneous Simultaneous Linear Equations is Solution
Let $S$ be a '''system of homogeneous simultaneous linear equations''': :$\displaystyle \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = 0$ Consider the trivial solution to $A$: :$\tuple {x_1, x_2, \ldots, x_n}$ such that: :$\forall j \in \set {1, 2, \ldots, n}: x_j = 0$ Then the trivial solution is indeed a solution to $S$.
Sine of Integer Multiple of Argument/Formulation 3
{{begin-eqn}} {{eqn | l = \sin n \theta       | r = \sin \theta \cos^{n - 1} \theta \paren {1 + 1 + \frac {\cos 2 \theta} {\cos^2 \theta} + \frac {\cos 3 \theta} {\cos^3 \theta} + \cdots + \frac {\cos \paren {n - 1} \theta} {\cos^{n - 1} \theta} }       | c =  }} {{eqn | r = \sin \theta \cos^{n - 1} \theta \sum_{k \mathop = 0}^{n - 1} \frac {\cos k \theta} {\cos^k \theta}       | c =  }} {{end-eqn}}
Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations
Let $S$ be a system of simultaneous linear equations: :$\displaystyle \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$ Let $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ denote the augmented matrix of $S$. Let $\begin {pmatrix} \mathbf A' & \mathbf b' \end {pmatrix}$ be obtained from $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ by means of an elementary row operation. Let $S'$ be the system of simultaneous linear equations of which $\begin {pmatrix} \mathbf A' & \mathbf b' \end {pmatrix}$ is the augmented matrix. Then $S$ and $S'$ are equivalent.
Existence of Inverse Elementary Row Operation
Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\map e {\mathbf A}$ be an elementary row operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$. Let $\map {e'} {\mathbf A'}$ be the inverse of $e$. Then $e'$ is an elementary row operation which always exists and is unique.
Superset of Dependent Set is Dependent
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $A, B \subseteq S$ such that $A \subseteq B$ If $A$ is a dependent subset then $B$ is a dependent subset.
Powers of 16 Modulo 20
Let $n \in \Z_{> 0}$ be a strictly positive integer. Then: :$16^n \equiv 16 \pmod {20}$
Sine of Integer Multiple of Argument/Formulation 1/Lemma
:For $n \in \Z$: {{begin-eqn}} {{eqn | l = \map \cos {n \theta} \map \sin {\theta}        | r = \map \sin {n \theta} \map \cos {\theta} - \map \sin {\paren {n - 1 } \theta}  }} {{end-eqn}}
Conjugacy Class of Identity is only Conjugacy Class which is Subgroup
Let $G$ be a group. Let $e$ denote the identity of $G$. Let $\conjclass g$ denote the conjugacy class of the element $g$. Then conjugacy class of identity is the only conjugacy class which is a subgroup of $G$: :$\conjclass g < G \iff g = e$
Empty Group Word is Reduced
Let $S$ be a set Let $\epsilon$ be the empty group word on $S$. Then $\epsilon$ is reduced.
Closed Unit Ball is Convex Set
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space. Let $\map {B_1^-} 0$ be a closed unit ball in $X$. Then $\map {B_1^-} 0$ is convex.
Uncountable Sum as Series/Corollary
Let $f: X \to \closedint 0 {+\infty}$ have uncountably infinite support. Then: :$\displaystyle \sum_{x \mathop \in X} \map f x = +\infty$
Identity Matrix from Upper Triangular Matrix
Let $\mathbf A = \sqbrk a_{m n}$ be an upper triangular matrix of order $m \times n$ with no zero diagonal elements. Let $k = \min \set {m, n}$. Then $\mathbf A$ can be transformed into a matrix such that the first $k$ rows and columns form the unit matrix of order $k$.
Simultaneous Linear Equations have Solution iff Ranks of Matrix of Coefficients and Augmented Matrix are Equal
Let $S$ be a system of simultaneous linear equations: :$\displaystyle \forall i \in \set {1, 2, \ldots, m} : \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$ Let $S$ be expressed in matrix form as: :$\mathbf A \mathbf x = \mathbf b$ where: :$\mathbf A = \begin {pmatrix} \alpha_{1 1} & \alpha_{1 2} & \cdots & \alpha_{1 n} \\ \alpha_{2 1} & \alpha_{2 2} & \cdots & \alpha_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ \alpha_{m 1} & \alpha_{m 2} & \cdots & \alpha_{m n} \\ \end {pmatrix}$,  $\mathbf x = \begin {pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}$, $\mathbf b = \begin {pmatrix} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_m \end {pmatrix}$ Then $S$ has at least one solution : :$\map \rho {\mathbf A} = \map \rho {\begin {array} {c|c} \mathbf A & \mathbf b \end {array} }$ where: :$\map \rho {\mathbf A}$ denotes the rank of $\mathbf A$ :$\paren {\begin {array} {c|c} \mathbf A & \mathbf b \end {array} }$ denotes the augmented matrix of $S$.
Simultaneous Linear Equations has Unique Solution iff Rank of Matrix of Coefficients equals Number of Columns
Let $S$ be a system of $m$ simultaneous linear equations in $n$ variables: :$\displaystyle \forall i \in \set {1, 2, \ldots, m} : \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$ Let $S$ be expressed in matrix form as: :$\mathbf A \mathbf x = \mathbf b$ where: :$\mathbf A = \begin {pmatrix} \alpha_{1 1} & \alpha_{1 2} & \cdots & \alpha_{1 n} \\ \alpha_{2 1} & \alpha_{2 2} & \cdots & \alpha_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ \alpha_{m 1} & \alpha_{m 2} & \cdots & \alpha_{m n} \\ \end {pmatrix}$,  $\mathbf x = \begin {pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}$, $\mathbf b = \begin {pmatrix} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_m \end {pmatrix}$ Then $S$ has exactly one solution : :$\map \rho {\mathbf A} = n$ where $\map \rho {\mathbf A}$ denotes the rank of $\mathbf A$.
Max Equals an Operand
Let $x_1, x_2, \dotsc, x_n \in S$ for some $n \in \N_{>0}$. Then: :$\exists i \in \closedint 1 n : x_i = \max \set {x_1, x_2, \dotsc, x_n}$
Similar Matrices have same Traces
Let $\mathbf A = \sqbrk a_n$ and $\mathbf B = \sqbrk b_n$ be square matrices of order $n$. Let $\mathbf A$ and $\mathbf B$ be similar. Then: :$\map \tr {\mathbf A} = \map \tr {\mathbf B}$ where $\map \tr {\mathbf A}$ denotes the trace of $\mathbf A$.
Cosine of Integer Multiple of Argument/Formulation 2
{{begin-eqn}} {{eqn | l = \cos n \theta       | r = \cos^n \theta \paren {1 - \dbinom n 2 \paren {\tan \theta}^2 + \dbinom n 4 \paren {\tan \theta}^4 - \cdots}       | c =  }} {{eqn | r = \cos^n \theta \sum_{k \mathop \ge 0} \paren {-1}^k \dbinom n {2 k } \paren {\tan^{2 k } \theta}       | c =  }} {{end-eqn}}
Polygamma Reflection Formula
:$\map {\psi_n} z - \paren {-1}^n \map {\psi_n} {1 - z} = -\pi \dfrac {\d^n} {\d z^n} \cot \pi z$
Polygamma Function in terms of Hurwitz Zeta Function
:$\displaystyle \map {\psi_n} z = \paren {-1}^{n + 1} \map \Gamma {n + 1} \map \zeta {n + 1, z}$
Area of Parallelogram from Determinant
Let $OABC$ be a parallelogram in the Cartesian plane whose vertices are located at: {{begin-eqn}} {{eqn | l = O       | r = \tuple {0, 0} }} {{eqn | l = A       | r = \tuple {a, c} }} {{eqn | l = B       | r = \tuple {a + b, c + d} }} {{eqn | l = C       | r = \tuple {b, d} }} {{end-eqn}} The area of $OABC$ is given by: :$\map \Area {OABC} = \begin {vmatrix} a & b \\ c & d \end {vmatrix}$ where $\begin {vmatrix} a & b \\ c & d \end {vmatrix}$ denotes the determinant of order $2$.
Matrix is Invertible iff Rank equals Order
Let $R$ be a commutative ring with unity. Let $\mathbf A \in R^{n \times n}$ be a square matrix of order $n$. Then $\mathbf A$ is invertible  its rank also equals $n$.
Determinant of Upper Triangular Matrix
Let $\mathbf T_n$ be an upper triangular matrix of order $n$. Let $\map \det {\mathbf T_n}$ be the determinant of $\mathbf T_n$. Then $\map \det {\mathbf T_n}$ is equal to the product of all the diagonal elements of $\mathbf T_n$. That is: :$\displaystyle \map \det {\mathbf T_n} = \prod_{k \mathop = 1}^n a_{k k}$
Cosine of Integer Multiple of Argument/Formulation 3
{{begin-eqn}} {{eqn | l = \cos n \theta       | r = \cos \paren {n - 1} \theta \cos \theta + \paren {1 - \sec^2 \theta } \cos^n \theta \paren {1 + 1 + \frac {\cos 2 \theta} {\cos^2 \theta} + \frac {\cos 3 \theta} {\cos^3 \theta} + \cdots + \frac {\cos \paren {n - 2} \theta} {\cos^{n - 2} \theta} }       | c =  }} {{eqn | r = \cos \paren {n - 1} \theta \cos \theta + \paren {1 - \sec^2 \theta } \cos^n \theta \sum_{k \mathop = 0}^{n - 2} \frac {\cos k \theta} {\cos^k \theta}       | c =  }} {{end-eqn}}
Determinant of Elementary Row Matrix/Scale Row
Let $e_1$ be the elementary row operation $\text {ERO} 1$: {{begin-axiom}} {{axiom | n = \text {ERO} 1         | t = For some $\lambda \ne 0$, multiply row $k$ by $\lambda$         | m = r_k \to \lambda r_k }} {{end-axiom}} which is to operate on some arbitrary matrix space. Let $\mathbf E_1$ be the elementary row matrix corresponding to $e_1$. The determinant of $\mathbf E_1$ is: :$\map \det {\mathbf E_1} = \lambda$
Determinant of Elementary Row Matrix/Scale Row and Add
Let $e_2$ be the elementary row operation $\text {ERO} 2$: {{begin-axiom}} {{axiom | n = \text {ERO} 2         | t = For some $\lambda$, add $\lambda$ times row $j$ to row $i$         | m = r_i \to r_i + \lambda r_j }} {{end-axiom}} which is to operate on some arbitrary matrix space. Let $\mathbf E_2$ be the elementary row matrix corresponding to $e_2$. The determinant of $\mathbf E_2$ is: :$\map \det {\mathbf E_2} = 1$
Determinant of Elementary Row Matrix/Exchange Rows
Let $e_3$ be the elementary row operation $\text {ERO} 3$: {{begin-axiom}} {{axiom | n = \text {ERO} 3         | t = Exchange rows $i$ and $j$         | m = r_i \leftrightarrow r_j }} {{end-axiom}} which is to operate on some arbitrary matrix space. Let $\mathbf E_3$ be the elementary row matrix corresponding to $e_3$. The determinant of $\mathbf E_3$ is: :$\map \det {\mathbf E_3} = -1$
Elementary Matrix corresponding to Elementary Column Operation
Let $\mathbf I$ denote the unit matrix of order $n$ over a field $K$. Let $e$ be an elementary column operation on $\mathbf I$. Let $\mathbf E$ be the elementary column matrix of order $n$ uniquely defined as: :$\mathbf E = e \paren {\mathbf I}$ where $\mathbf I$ is the unit matrix. Let $\kappa_k$ denote the $k$th column of $\mathbf I$ for $1 \le k \le n$.
Elementary Matrix corresponding to Elementary Column Operation/Scale Column
Let $e$ be the elementary column operation acting on $\mathbf I$ as: {{begin-axiom}} {{axiom | n = \text {ECO} 1         | t = For some $\lambda \in K_{\ne 0}$, multiply column $k$ of $\mathbf I$ by $\lambda$         | m = \kappa_k \to \lambda \kappa_k }} {{end-axiom}}
Elementary Matrix corresponding to Elementary Column Operation/Scale Column and Add
Let $e$ be the elementary column operation acting on $\mathbf I$ as: {{begin-axiom}} {{axiom | n = \text {ECO} 2         | t = For some $\lambda \in K$, add $\lambda$ times column $j$ to row $i$         | m = \kappa_i \to \kappa_i + \lambda r_j }} {{end-axiom}}
Elementary Matrix corresponding to Elementary Column Operation/Exchange Columns
Let $e$ be the elementary column operation acting on $\mathbf I$ as: {{begin-axiom}} {{axiom | n = \text {ECO} 3         | t = Interchange columns $i$ and $j$         | m = \kappa_i \leftrightarrow \kappa_j }} {{end-axiom}}
Row Operation has Inverse
Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\Gamma$ be a row operation which transforms $\mathbf A$ to a new matrix $\mathbf B \in \map \MM {m, n}$. Then there exists another row operation $\Gamma'$ which transforms $\mathbf B$ back to $\mathbf A$.
Real Numbers with Absolute Value form Normed Vector Space
Let $\R$ be the set of real numbers. Let $\size {\, \cdot \,}$ be the absolute value. Then $\struct {\R, \size {\, \cdot \,}}$ is a normed vector space.
Row Operation is Equivalent to Pre-Multiplication by Product of Elementary Matrices
Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\Gamma$ be a row operation which transforms $\mathbf A$ to a new matrix $\mathbf B \in \map \MM {m, n}$. Then there exists a unique invertible square matrix $\mathbf R$ of order $m$ such that: :$\mathbf R \mathbf A = \mathbf B$ where $\mathbf R$ is the product of a finite sequence of elementary row matrices.
Product of Matrices is Invertible iff Matrices are Invertible
Let $\mathbf A$ and $\mathbf B$ be square matrices of order $n$. Let $\mathbf A \mathbf B$ denote the matrix product of $\mathbf A$ and $\mathbf B$. Let $\mathbf I$ be the $n \times n$ unit matrix. Let $\mathbf A$ and $\mathbf B$ be invertible. Then: :$\mathbf A \mathbf B$ is invertible  :both $\mathbf A$ and $\mathbf B$ are invertible.
Square Root of Number Plus or Minus Square Root
Let $a$ and $b$ be (strictly) positive real numbers such that $a^2 - b > 0$. Then:
Elementary Row Matrix is Invertible
Let $\mathbf E$ be an elementary row matrix. Then $\mathbf E$ is invertible.
Existence of Inverse Elementary Column Operation
Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\map e {\mathbf A}$ be an elementary column operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$. Let $\map {e'} {\mathbf A'}$ be the inverse of $e$. Then $e'$ is an elementary column operation which always exists and is unique.
Column Operation has Inverse
Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\Gamma$ be a column operation which transforms $\mathbf A$ to a new matrix $\mathbf B \in \map \MM {m, n}$. Then there exists another column operation $\Gamma'$ which transforms $\mathbf B$ back to $\mathbf A$.
Elementary Column Matrix is Invertible
Let $\mathbf E$ be an elementary column matrix. Then $\mathbf E$ is invertible.