# 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: $E_{1}=\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ -2 & 0 & 1 \end{bmatrix}$, $E_{2}=\begin{bmatrix} 0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1 \end{bmatrix}$, $E_{3}=\begin{bmatrix} 1 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 1 \end{bmatrix}$, and $A=\begin{bmatrix} x_{11} & x_{12} & x_{13}\\ x_{21} & x_{22} & x_{23}\\ x_{31} & x_{32} & x_{33} \end{bmatrix}$. $E_{1}$, $E_{2}$ 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: $E_{1}=\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & -3 & 1 \end{bmatrix}$, $E_{2}=\begin{bmatrix} 1 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0 \end{bmatrix}$, $E_{3}=\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 3 \end{bmatrix}$, and $A=\begin{bmatrix} x_{11} & x_{12} & x_{13}\\ x_{21} & x_{22} & x_{23}\\ x_{31} & x_{32} & x_{33} \end{bmatrix}$. $E_{1}$, $E_{2}$ 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 $E_{i}'s$ are elementary matrices that transform $A$ into $I_{n}$.

Example 2: Find $A^{-1}$ where $A=\begin{bmatrix} 1 & 0 & 3\\ 0 & 1 & 1\\ 2 & -1 & 2 \end{bmatrix}$.

Exercise 2: Find $A^{-1}$ where $A=\begin{bmatrix} 2 & 1 & 0\\ 1 & 1 & 2\\ 0 & -1 & 2 \end{bmatrix}$.

Remark: Let $A$ be an invertible matrix and $I = \begin{bmatrix} \vec{e_{1}} & \cdots & \vec{e_{n}} \end{bmatrix}$ then there are $\vec{u_{1}} \cdots \vec{u_{n}}$ such that $A\vec{u_{i}} = \vec{e_{i}}$ for $i = 1, \cdots, n$. The augmented matrix $\begin{bmatrix} A\vec{e_{i}} \end{bmatrix}$ will be equivalent to $\begin{bmatrix} I_{n}\vec{u_{1}} \end{bmatrix}$ by using the same elementary row operations, $E_{p}, \cdots, E_{1}$ for $i = 1, \cdots, n$. Therefore we can write $\begin{bmatrix} A | \vec{e_{1}} \cdots \vec{e_{n}} \end{bmatrix} = \begin{bmatrix} A | I_{n} \end{bmatrix}$ is equivalent to $\begin{bmatrix} I_{n} | \vec{u_{1}} \cdots \vec{u_{n}} \end{bmatrix}$. From the theorem and the example above, we know $\begin{bmatrix} \vec{u_{1}} & \cdots & \vec{u_{n}} \end{bmatrix}$ is the inverse matrix of $A$, i.e. $A^{-1} = \begin{bmatrix} \vec{u_{1}} & \cdots & \vec{u_{n}} \end{bmatrix}$, the solutions of $A\vec{x} = \vec{e_i}$ 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 $A=\begin{bmatrix} 3 & 4 & -3\\ 0 & 1 & 2\\ -1 & 0 & 4 \end{bmatrix}$.

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

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 $\begin{bmatrix}A|I_{m}\end{bmatrix} \rightarrow \begin{bmatrix}B|U\end{bmatrix}$ 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 $A=\begin{bmatrix} 1 & 3 & 2\\ 2 & 1 & -1 \end{bmatrix}$, express the reduced row-echelon form $R$ of $A$ as $R = UA$ where $U$ is invertible.

Exercise 4: If $A=\begin{bmatrix} 0 & 3 & 1\\ 1 & 1 & 1 \end{bmatrix}$, 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 $A=\begin{bmatrix} 1 & 3\\ 2 & 1 \end{bmatrix}$ as product of elementary matrices.

Exercise 5: Express $A=\begin{bmatrix} -1 & 1\\ 1 & 2 \end{bmatrix}$ as product of elementary matrices.

Group Work 1: Mark each statement True or False. Justify each answer.

a. If $A$ is an invertible $n \times n$ matrix then the equation $A\vec{x} = \vec{b}$ is consistent for each $\vec{b}$ 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 $A\vec{x} = \vec{e_{i}}$ is consistent for $i = 1, \cdots, n$ and $\vec{e_{i}}$ is the i-th column of $I_{n}$. Then $A$ is invertible.

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

Group Work 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 $B\vec{x} = 0$ has a non-trivial solution.

Group Work 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. $\begin{bmatrix}P|Q\end{bmatrix}$ is obtained by row operations from $\begin{bmatrix}A|I\end{bmatrix}$ then $P=QA$.

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