Definition and Theorems of Chapter 1

Theorem 1. Equivalent systems of linear equations have the same solutions.

Theorem 2. To each elementary row operations $\bold{e}$ there corresponds an elementary row operation $\bold{e}_1$, of the same type as $\bold{e}$, such that $\bold{e}_1(\bold{e}(A))=\bold{e}(\bold{e}_1(A))=A$ for each $A$. In other words, the inverse operation (function) of an elementary row operation exists and is an elementary row operation of the same type.

Definition. If $A$ and $B$ are $m\times n$ matrices over the field $F$, we say that $B$ is row-equivalent to $A$ if $B$ can be obtained from $A$ by a finite sequence of elementary row operations.

Theorem 3. If $A$ and $B$ are row-equivalent $m\times n$ matrices, the homogeneous systems of linear equations $AX=0$ and $BX=0$ have exactly the same solutions.

Definition. An $m\times n$ matrix $R$ is called row-reduced if:
( a ) the first non-zero entry in each non-zero row of $R$ is equal to 1;
( b ) each column of $R$ which contains the leading non-zero entry of some row has all its other entries 0.

Theorem 4. Every $m\times n$ matrix over the field $F$ is row-equivalent to a row-reduced matrix.

Definition. An $m\times n$ matrix $R$ is called a row-reduced echelon matrix if:
( a ) $R$ is row-reduced;
( b ) every row of $R$ which has all its entries 0 occurs below every row which has a non-zero entry;
( c ) if rows $1,\dots,r$ are the non-zero rows of $R$, and if the leading non-zero entry of row $i$ occurs in column $k_i,i=1,\dots,r$, then $k_1<k_2<...<k_r$.

Theorem 5. Every $m\times n$ matrix $A$ is row-equivalent to a row-reduced echelon matrix.

Theorem 6. If $A$ is an $m\times n$ matrix and $m<n$, then the homogeneous system of linear equations $AX=0$ has a non-trivial solution.

Theorem 7. If $A$ is an $n\times n$ (square) matrix, then $A$ is row-equivalent to the $n\times n$ identity matrix if and only if the system of equations $AX=0$ has only the trivial solution.

Definition. Let $A$ be an $m\times n$ matrix over the field $F$ and let $B$ be an $n\times p$ matrix over $F$. The product $AB$ is the $m\times p$ matrix $C$ whose $i,j$ entry is $C_{ij}=\sum_{r=1}^{n}A_{ir}B_{rj}$.

Theorem 8. If $A,B,C$ are matrices over the field $F$ such that the products $BC$ and $A(BC)$ are defined, then so are the products $AB,(AB)C$ and
$A(BC)=(AB)C$

Definition. An $m\times n$ matrix is said to be an elementary matrix if it can be obtained from the $m\times m$ identity matrix by emans of a single elementary row operation.

Theorem 9. Let $e$ be an elementary row operation and let $E$ be the $m\times m$ elementary matrix $E=e(I)$. Then for every $m\times n$ matrix $A$, $e(A)=EA$.
Corollary. Let $A$ and $B$ be $m\times n$ matrices over the field $F$. Then $B$ is row-equivalent to $A$ if and only if $B=PA$, where $P$ is a product of $m\times m$ elementary matrices.

Definition. Let $A$ be an $n\times n$ (square) matrix over the field $F$. An $n\times n$ matrix $B$ such that $BA=I$ is called a left inverse of $A$; an $n\times n$ matrix $B$ such that $AB=I$ is called a right inverse of $A$. IF $AB=BA=I$, then $B$ is called a two-sided inverse (or inverse) of $A$ and $A$ is said to be invertible.

Theorem 10. Let $A$ and $B$ be $n\times n$ matrices over $F$.
( i ) If $A$ is invertible, so is $A^{-1}$ and $(A^{-1})^{-1}=A$.
( ii ) If both $A$ and $B$ are invertible, so is $AB$ and $(AB)^{-1}=B^{-1}A^{-1}$.
Corollary. A product of invertible matrices is invertible.

Theorem 11. An elementary matrix is invertible.

Theorem 12. If $A$ is an $n\times n$ matrix, the following are equivalent.
( i ) $A$ is invertible.
( ii ) $A$ is row-equivalent to the $n\times n$ identity matrix.
( iii ) $A$ is a product of elementary matrices.
Corollary. If $A$ is an invertible $n\times n$ matrix and if a sequence of elementary row operations reduces $A$ to the identity, then that same sequence of operations when applied to $I$ yields $A^{-1}$.
Corollary. Let $A$ and $B$ be $m\times n$ matrces. Then $B$ is row-equivalent to $A$ if and only if $B=PA$ where $P$ is an invertible $m\times m$ matrix.

Theorem 13. For an $n\times n$ matrix $A$, the following are equivalent.
( i ) $A$ is invertible.
( ii ) The homogeneous system $AX=0$ has only the trivial solution $X=0$.
( iii ) The system of equations $AX=Y$ has a solution $X$ for each $n\times 1$ matrix $Y$.
Corollary. A square matrix with either a left or right inverse is invertible.
Corollary. Let $A=A_1A_2{\cdots}A_k$ where $A_1,\dots,A_k$ are $n\times n$ (square) matrices. Then $A$ is invertible if and only if each $A_j$ is invertible.