Section 2.5 Elementary Matrices

Definition: An elementary matrix is one that is obtained by performing a single elementary row operation on an identity matrix.

 

Example 1: $E_1=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ -2& 0& 1\end{array}\right]$, $E_2=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$, $E_3=\left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\\ 0& 0& 1\end{array}\right]$,

and $A=\left[\begin{array}{ccc}{x}_{11}& x_{12}& x_{13}\\ x_{21}& x_{22}& x_{23}\\ x_{31}& x_{32}& x_{33}\end{array}\right]$. $E_1$, $E_2$ and $E_3$ are elementary matrices.

Describe how to get $E_1$, $E_2$ and $E_3$ from identity matrix $I_3$ by elementary row operations. Compute $E_{1}A$, $E_{2}A$ and $E_{3}A$ and describe how these products can be obtained by elementary row operations.

 

 

Exercise 1: $E_1=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& -3& 1\end{array}\right]$, $E_2=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]$, $E_3=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 3\end{array}\right]$,

and $A=\left[\begin{array}{ccc}{x}_{11}& x_{12}& x_{13}\\ x_{21}& x_{22}& x_{23}\\ x_{31}& x_{32}& x_{33}\end{array}\right]$. $E_1$, $E_2$ and $E_3$ are elementary matrices.

Describe how to get $E_1$, $E_2$ and $E_3$ from identity matrix $I_3$ by elementary row operations. Compute $E_{1}A$, $E_{2}A$ and $E_{3}A$ and describe how these products can be obtained by elementary row operations.

 

Fact: 1. If an elementary row operation is performed on an $m\times n$ matrix $A$, the resulting matrix can be written as $EA$, where the $m\times m$ matrix $E$ is created by performing the same row operation on $I_m$.

 

2. Each elementary matrix is invertible. The inverse of $E$ is the elementary matrix of the same type that transforms $E$ back into $I$.

 

Theorem: An $n\times n$ matrix is invertible if and only if $A$ is row equivalent to $I_n$ and in this case, any sequence of elementary row operations that reduces $A$ to $I_n$ also transforms $I_n$ into $A^{-1}$.

 

Proof:

 

Fact: If $E_p{E}_{p-1}\cdots E_{1}A=I_n$ then $A^{-1}=(E_p\cdots E_1)I_n$ where $E_i^{\prime }s$ are elementary matrices that transform $A$ into $I_n$.

 

 

Example 2: Find $A^{-1}$ where $A=\left[\begin{array}{ccc}1& 0& 3\\ 0& 1& 1\\ 2& -1& 2\end{array}\right]$.

 

 

Exercise 2: Find $A^{-1}$ where $A=\left[\begin{array}{ccc}2& 1& 0\\ 1& 1& 2\\ 0& -1& 2\end{array}\right]$.

 

Remark: Let $A$ be an invertible matrix and $I=\left[\begin{array}{ccc}\overrightarrow{e_1}& \cdots & \overrightarrow{e_n}\end{array}\right]$ then there are $\overrightarrow{u_1}\cdots \overrightarrow{u_n}$ such that $A\overrightarrow{u_i}=\overrightarrow{e_i}$ for $i=1,\cdots ,n$. The augmented matrix $\left[\begin{array}{c}A\overrightarrow{e_i}\end{array}\right]$ will be equivalent to $\left[\begin{array}{c}{I}_n\overrightarrow{u_1}\end{array}\right]$ by using the same elementary row operations, $E_p,\cdots ,E_1$ for $i=1,\cdots ,n$. Therefore we can write $\left[\begin{array}{c}A|\overrightarrow{e_1}\cdots \overrightarrow{e_n}\end{array}\right]=\left[\begin{array}{c}A|I_n\end{array}\right]$ is equivalent to $\left[\begin{array}{c}{I}_n|\overrightarrow{u_1}\cdots \overrightarrow{u_n}\end{array}\right]$. From the theorem and the example above, we know $\left[\begin{array}{ccc}\overrightarrow{u_1}& \cdots & \overrightarrow{u_n}\end{array}\right]$ is the inverse matrix of $A$, i.e. $A^{-1}=\left[\begin{array}{ccc}\overrightarrow{u_1}& \cdots & \overrightarrow{u_n}\end{array}\right]$, the solutions of $A\overrightarrow{x}=\overrightarrow{e_i}$ for $i=1,\cdots ,n$ form the columns of $A^{-1}$.

 

Example 3: Find the third column of $A^{-1}$ without computing the other columns, where $A=\left[\begin{array}{ccc}3& 4& -3\\ 0& 1& 2\\ -1& 0& 4\end{array}\right]$.

 

 

Exercise 3: Find the third column of $A^{-1}$ without computing the other columns, where $A=\left[\begin{array}{ccc}1& 4& 3\\ -2& 0& 2\\ 0& -1& 5\end{array}\right]$.

 

Theorem: Suppose $A$ is $m\times n$ and $B$ is obtained by $A$ by elementary row operations.

1. $B=UA$ where $U$ is an $m\times m$ invertible matrix

2. $U$ can be computed by $\left[\begin{array}{c}A|I_m\end{array}\right]\to \left[\begin{array}{c}B|U\end{array}\right]$ using the operations carrying $A$ to $B$.

3. Where $E_1,\cdots ,E_k$ are the elementary matrices corresponding (in order) to the elementary row operations carrying $A$ to $B$.

 

Example 4: If $A=\left[\begin{array}{ccc}1& 3& 2\\ 2& 1& -1\end{array}\right]$, express the reduced row-echelon form $R$ of $A$ as $R=UA$ where $U$ is invertible.

 

 

Exercise 4: If $A=\left[\begin{array}{ccc}0& 3& 1\\ 1& 1& 1\end{array}\right]$, express the reduced row-echelon form $R$ of $A$ as $R=UA$ where $U$ is invertible.

 

Theorem: A square matrix is invertible if and only if it is a product of elementary matrices.

 

Example 5: Express $A=\left[\begin{array}{cc}1& 3\\ 2& 1\end{array}\right]$ as product of elementary matrices.

 

 

Exercise 5: Express $A=\left[\begin{array}{cc}-1& 1\\ 1& 2\end{array}\right]$ as product of elementary matrices.

 

GroupWork 1: Mark each statement True or False. Justify each answer.

a. If $A$ is an invertible $n\times n$ matrix then the equation $A\overrightarrow{x}=\overrightarrow{b}$ is consistent for each $\overrightarrow{b}$ in ${\mathbb{R}}^n$.

 

b. Each elementary matrix is invertible.

 

c. If $A$ is invertible, then elementary row operations that reduce $A$ to the identity $I_n$ also reduce $A^{-1}$ to $I_n$.

 

d. If $A$ is invertible then the inverse of $A^{-1}$ is $A$.

 

e. $A$ is an $n\times n$ matrix and $A\overrightarrow{x}=\overrightarrow{e_i}$ is consistent for $i=1,\cdots ,n$ and $\overrightarrow{e_i}$ is the i-th column of $I_n$. Then $A$ is invertible.

 

GroupWork 2: Let $E$ be an elementary matrix. Show that $E^T$ is also an elementary matrix.

 

GroupWork 3: Let $A$ and $B$ be $m\times n$ and $n\times m$ matrices, respectively. If $m>n$, show that $AB$ is not invertible. Hint: Use $B\overrightarrow{x}=0$ has a non-trivial solution.

 

GroupWork 4: Mark each statement True or False. Justify each answer.

a. If $A$ can be row reduced to identity matrix then $A$ is invertible.

 

b. $0$ an elementary matrix.

 

c. $I$ an elementary matrix.

 

d. $\left[\begin{array}{c}P|Q\end{array}\right]$ is obtained by row operations from $\left[\begin{array}{c}A|I\end{array}\right]$ then $P=QA$.

 

e. If $A\overrightarrow{x}=0$ has trivial solution only then $A$ is a product of elementary matrices.