Question

# Let A=begin{bmatrix}1 & 2 -1 & 1 end{bmatrix} text{ and } C=begin{bmatrix}-1 & 1 2 & 1 end{bmatrix}a)Find elementary matrices E_1 text{ and } E_2 such that C=E_2E_1Ab)Show that is no elementary matrix E such that C=EA

Matrices

Let $$A=\begin{bmatrix}1 & 2 \\-1 & 1 \end{bmatrix} \text{ and } C=\begin{bmatrix}-1 & 1 \\2 & 1 \end{bmatrix}$$
a)Find elementary matrices $$E_1 \text{ and } E_2$$ such that $$C=E_2E_1A$$
b)Show that is no elementary matrix E such that $$C=EA$$

2020-12-08

Step 1
Consider the given information,
$$A=\begin{bmatrix}1 & 2 \\-1 & 1 \end{bmatrix} \text{ and } C=\begin{bmatrix}-1 & 1 \\2 & 1 \end{bmatrix}$$
a)
now, calculate the elementary matrices.
The matrix C can be be calculated by A by the following operations.
Step 2
Interchange the Rows of the matrix A. Then,
$$A^*=\begin{bmatrix}-1 & 1 \\1 & 2 \end{bmatrix}$$
Now, multiply -1 in the row one and add in second.
$$A^*=\begin{bmatrix}-1 & 1 \\2 & 1 \end{bmatrix}$$
Then the elementary matrix are defined as,
$$E_1=\begin{bmatrix}0 & 1 \\1 & 0 \end{bmatrix}$$
And
$$E_2=\begin{bmatrix}1 & 0 \\-1 & 1 \end{bmatrix}$$
Step 3
(b) It can be observed from part a. A and C lines above are equivalent. However, none of the 2 matrices A and C can be changed to another by a single row operation. Hence there is no primary matrix E such that $$C = EA.$$