# 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

Question
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.

### Relevant Questions

Consider the matrices
$$A=\begin{bmatrix}1 & -1 \\0 & 1 \end{bmatrix},B=\begin{bmatrix}2 & 3 \\1 & 5 \end{bmatrix},C=\begin{bmatrix}1 & 0 \\0 & 8 \end{bmatrix},D=\begin{bmatrix}2 & 0 &-1\\1 & 4&3\\5&4&2 \end{bmatrix} \text{ and } F=\begin{bmatrix}2 & -1 &0\\0 & 1&1\\2&0&3 \end{bmatrix}$$
a) Show that A,B,C,D and F are invertible matrices.
b) Solve the following equations for the unknown matrix X.
(i) $$AX^T=BC^3$$
(ii) $$A^{-1}(X-T)^T=(B^{-1})^T$$
(iii) $$XF=F^{-1}-D^T$$
Given the matrix
$$A=\begin{bmatrix}0 & 0&1 \\ 0 & 3&0 \\ 1&0 & -10 \end{bmatrix}$$
and suppose that we have the following row reduction to its PREF B
$$A=\begin{bmatrix}0 & 0&1 \\ 0 & 3&0 \\ 1&0 & -10 \end{bmatrix}\Rightarrow\begin{bmatrix}1 & 0&-10 \\ 0 & 3&0 \\ 0&0 & 1 \end{bmatrix}\Rightarrow\begin{bmatrix}1 & 0&-10 \\ 0 & 1&0 \\ 0&0 & 1 \end{bmatrix}\Rightarrow\begin{bmatrix}1 & 0&0 \\ 0 & 1&0 \\ 0&0 & 1 \end{bmatrix}$$
Write $$A \text{ and } A^{-1}$$ as product of elementary matrices.
Which of the following matrices is elementary matrix?
a) $$\begin{bmatrix}0 & 3 \\1 & 0 \end{bmatrix}$$
b) $$\begin{bmatrix}2 & 0 \\0 & 1 \end{bmatrix}$$
c) $$\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix}$$
d) $$\begin{bmatrix}2 & 0 \\0 & 2 \end{bmatrix}$$
Given the matrices
$$A=\begin{bmatrix}5 & 3 \\ -3 & -1 \\ -2 & -5 \end{bmatrix} \text{ and } B=\begin{bmatrix}0 & -2 \\ 1 & 3 \\ 4 & -3 \end{bmatrix}$$
find the 3x2 matrix X that is a solution of the equation. 2X-A=X+B
X=?
Given the matrices
$$A=\begin{bmatrix}-1 & 3 \\2 & -1 \\ 3&1 \end{bmatrix} \text{ and } B=\begin{bmatrix}0 & -2 \\1 & 3 \\ 4 & -3 \end{bmatrix}$$ find the $$3 \times 2$$ matrix X that is a solution of the equation. 8X+A=B
Let M be the vector space of $$2 \times 2$$ real-valued matrices.
$$M=\begin{bmatrix}a & b \\c & d \end{bmatrix}$$
and define $$M^{#}=\begin{bmatrix}d & b \\c & a \end{bmatrix}$$ Characterize the matrices M such that $$M^{#}=M^{-1}$$
Let $$u=\begin{bmatrix}2 \\ 5 \\ -1 \end{bmatrix} , v=\begin{bmatrix}4 \\ 1 \\ 3 \end{bmatrix} \text{ and } w=\begin{bmatrix}-4 \\ 17 \\ -13 \end{bmatrix}$$ It can be shown that 4u-3v-w=0. Use this fact (and no row operations) to find a solution to the system Ax=b , where
$$A=\begin{bmatrix}2 & -4 \\5 & 17\\-1&-13 \end{bmatrix} , x=\begin{bmatrix}x_1 \\ x_2 \end{bmatrix} , b=\begin{bmatrix}4 \\ 1 \\ 3 \end{bmatrix}$$
$$P=\begin{bmatrix}-1 & -1 \\1& 2 \end{bmatrix}, A=\begin{bmatrix}14 & 9 \\-20 & -13 \end{bmatrix} \text{ and } A'=\begin{bmatrix}3 & -2 \\2 & -2 \end{bmatrix}$$
$$P^{-1}=?$$
$$P^{-1}AP=?$$
Let $$A=\begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix} \text{ and } B=\begin{bmatrix}1 & 2 \\ 0 & 1 \end{bmatrix}$$ , show that A and B are not similar.
Let $$A=\begin{bmatrix}3 & 0 \\ -1 & 5 \end{bmatrix} , B=\begin{bmatrix}4 & -2 & 1 \\ 0 & 2 &3 \end{bmatrix} , C=\begin{bmatrix}1 & 2 \\ 3 & 4 \\ 5 &6 \end{bmatrix} , D=\begin{bmatrix}0 & -3 \\ -2 & 1 \end{bmatrix} , E=\begin{bmatrix}4 & 2 \end{bmatrix} , F=\begin{bmatrix}-1 \\ 2 \end{bmatrix}$$