Elementary Matrices

DEFINITION 2.3.13   A square matrix $E$ of order $n$ is called an elementary matrix if it is obtained by applying exactly one elementary row operation to the identity matrix, $I_n.$

Remark 2.3.14   There are three types of elementary matrices.

1. $E_{ij},$ which is obtained by the application of the elementary row operation $R_{ij}$ to the identity matrix, $I_n.$ Thus, the $(k, \ell)^{\mbox{th}}$ entry of $E_{ij}$ is $$(E_{ij})_{(k,\ell)} = \begin{cases}1 & {\mbox{ if }} k=\ell {... ...or }} (k,\ell) = (j,i) \\ 0 & {\mbox{ otherwise}} \end{cases}.$$
2. $E_k(c),$ which is obtained by the application of the elementary row operation $R_k(c)$ to the identity matrix, $I_n.$ The $(i,j)^{\mbox{th}}$ entry of $E_k(c)$ is $(E_k(c))_{(i,j)}= \begin{cases}1 & {\mbox{ if }} i=j {\mbox{ and }} i \neq k \\ c & {\mbox{ if }} i=j = k \\ 0 & {\mbox{ otherwise}} \end{cases}.$
3. $E_{ij}(c),$ which is obtained by the application of the elementary row operation $R_{ij}(c)$ to the identity matrix, $I_n.$ The $(k, \ell)^{\mbox{th}}$ entry of $E_{ij}(c)$ is $(E_{ij})_{(k,\ell)}\begin{cases}1 & {\mbox{ if }} k=\ell \\ c & {\mbox{ if }} (k,\ell) = (i,j) \\ 0 & {\mbox{ otherwise}} \end{cases}.$

In particular, if we start with a $3 \times 3$ identity matrix $I_3$ , then

$$E_{23} = \begin{bmatrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{... ... E_{23}(c) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & c \\ 0 & 0 & 1 \end{bmatrix}.$$

EXAMPLE 2.3.15  

1. Let $A = \begin{bmatrix}1 & 2 & 3 & 0 \\ 2 & 0 & 3 & 4\\ 3 & 4 & 5 & 6 \end{bmatrix}.$ Then
   $$\begin{bmatrix}1 & 2 & 3 & 0 \\ 2 & 0 & 3 & 4\\ 3 & 4 & 5 & 6 \en... ... \begin{bmatrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix} A = E_{23} A.$$
   That is, interchanging the two rows of the matrix $A$ is same as multiplying on the left by the corresponding elementary matrix. In other words, we see that the left multiplication of elementary matrices to a matrix results in elementary row operations.
2. Consider the augmented matrix $[A \; {\mathbf b}] = \begin{bmatrix}0 & 1 & 1 & 2 \\ 2 & 0 & 3 & 5 \\ 1 & 1 & 1 & 3 \end{bmatrix}.$ Then the result of the steps given below is same as the matrix product
   $$E_{23}(-1) E_{12}(-1) E_3(1/3) E_{32}(2) E_{23} E_{21}(-2) E_{13} [A \; {\mathbf b}].$$
   
   

   $$\begin{bmatrix}0 & 1 & 1 & 2 \\ 2 & 0 & 3 & 5 \\ 1 & 1 & 1 & 3 \end{bmatrix} \overrightarrow{R_{13} } \begin{bmatrix}1 & 1 & 1 & 3 \\ 2 & 0 & 3 & 5 \\ 0 & 1 & 1 & 2 \e... ... \begin{bmatrix}1 & 1 & 1 & 3 \\ 0 & 1 & 1 & 2 \\ 0 & -2 & 1 & -1 \end{bmatrix}$$

   $$\overrightarrow{R_{32}(2)} \begin{bmatrix}1 & 1 & 1 & 3 \\ 0 & 1 & 1 & 2 \\ 0 & 0 & 3 & 3 \e... ...)} \begin{bmatrix}1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 2 \\ 0 & 0 & 1 & 1 \end{bmatrix}$$

   $$\overrightarrow{R_{23}(-1) } \begin{bmatrix}1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \end{bmatrix}$$

   
   

Now, consider an $m \times n$ matrix $A$ and an elementary matrix $E$ of order $n.$ Then multiplying by $E$ on the right to $A$ corresponds to applying column transformation on the matrix $A.$ Therefore, for each elementary matrix, there is a corresponding column transformation. We summarize:

DEFINITION 2.3.16   The column transformations obtained by right multiplication of elementary matrices are called elementary column operations.

EXAMPLE 2.3.17   Let $A = \begin{bmatrix}1 & 2 & 3 \\ 2 & 0 & 3 \\ 3 & 4 & 5 \end{bmatrix}$ and consider the elementary column operation ${\mathit f}$ which interchanges the second and the third column of $A.$ Then $f(A) = \begin{bmatrix}1 & 3 & 2 \\ 2 & 3 & 0 \\ 3 & 5 & 4 \end{bmatrix} = A \begin{bmatrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix} = A E_{23}.$

EXERCISE 2.3.18  

1. Let ${\mathit e}$ be an elementary row operation and let $E = e(I)$ be the corresponding elementary matrix. That is, $E$ is the matrix obtained from $I$ by applying the elementary row operation $e.$ Show that $e(A) = E A.$
2. Show that the Gauss elimination method is same as multiplying by a series of elementary matrices on the left to the augmented matrix. Does the Gauss-Jordan method also corresponds to multiplying by elementary matrices on the left? Give reasons.
3. Let $A$ and $B$ be two $m \times n$ matrices. Then prove that the two matrices $A, B$ are row-equivalent if and only if $B = P A,$ where $P$ is product of elementary matrices. When is this $P$ unique?
4. Show that every elementary matrix is invertible. Is the inverse of an elementary matrix, also an elementary matrix?