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
asked 2020-12-07
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

Answers (1)

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.
0

Relevant Questions

asked 2020-12-16
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) \(XF=F^{-1}-D^T\)
asked 2021-01-22
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.
asked 2020-11-08
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) \(\begin{bmatrix}2 & 0 \\0 & 2 \end{bmatrix}\)
asked 2021-01-30
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=?
asked 2021-02-02
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
asked 2020-11-30
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 \(M^{#}=M^{-1}\)
asked 2021-02-24
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
\(A=\begin{bmatrix}2 & -4 \\5 & 17\\-1&-13 \end{bmatrix} , x=\begin{bmatrix}x_1 \\ x_2 \end{bmatrix} , b=\begin{bmatrix}4 \\ 1 \\ 3 \end{bmatrix}\)
asked 2020-10-26
Use the matrix P to determine if the matrices A and A' are similar.
\(P=\begin{bmatrix}-1 & -1 \\1& 2 \end{bmatrix}, A=\begin{bmatrix}14 & 9 \\-20 & -13 \end{bmatrix} \text{ and } A'=\begin{bmatrix}3 & -2 \\2 & -2 \end{bmatrix}\)
\(P^{-1}=?\)
\(P^{-1}AP=?\)
Are they similar?
"Yes, they are similar" or "No, they are not similar"
asked 2020-11-08
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.
asked 2021-02-02
compute the indicated matrices (if possible). D+BC
Let \(A=\begin{bmatrix}3 & 0 \\ -1 & 5 \end{bmatrix} , B=\begin{bmatrix}4 & -2 & 1 \\ 0 & 2 &3 \end{bmatrix} , C=\begin{bmatrix}1 & 2 \\ 3 & 4 \\ 5 &6 \end{bmatrix} , D=\begin{bmatrix}0 & -3 \\ -2 & 1 \end{bmatrix} , E=\begin{bmatrix}4 & 2 \end{bmatrix} , F=\begin{bmatrix}-1 \\ 2 \end{bmatrix}\)
...