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

Question

Let $A = [\begin{matrix} 1 & 2 \\ - 1 & 1 \end{matrix}] and C = [\begin{matrix} - 1 & 1 \\ 2 & 1 \end{matrix}]$
a)Find elementary matrices $E_{1} and E_{2}$ such that $C = E_{2} E_{1} A$
b)Show that is no elementary matrix E such that $C = E A$

Answer 1

Step 1
Consider the given information,
$A = [\begin{matrix} 1 & 2 \\ - 1 & 1 \end{matrix}] and C = [\begin{matrix} - 1 & 1 \\ 2 & 1 \end{matrix}]$
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,
$A^{*} = [\begin{matrix} - 1 & 1 \\ 1 & 2 \end{matrix}]$
Now, multiply -1 in the row one and add in second.
$A^{*} = [\begin{matrix} - 1 & 1 \\ 2 & 1 \end{matrix}]$
Then the elementary matrix are defined as,
$E_{1} = [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]$
And
$E_{2} = [\begin{matrix} 1 & 0 \\ - 1 & 1 \end{matrix}]$
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 $C = E A .$