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
ANSWERED
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 \(C = EA.\)

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