Question

Consider the matrices A = −2 7 1

3 4 1

8 1 5 ,

B = 8 1 5

3 4 1

−2 7 1 ,

C = −2 7 1

3 4 1

2 −7 3 .

Find elementary matrices E1, E2 and E3 such that

(5.1) E1A = B,

(5.2) E1B = A,

(5.3) E2A = C,

(5.4) E3C = A.

Answer 1

$A=\begin{pmatrix} −2 & 7 & 1 \\ 3 & 4 & 1 \\ 8 & 1 & 5 \end{pmatrix} B=\begin{pmatrix} 8 & 1 & 5 \\ 3 & 4 & 1 \\ −2 & 7 & 1 \end{pmatrix} C=\begin{pmatrix} −2 & 7 & 1\\ 3 & 4 & 1\\ 2 & −7 & 3 \end{pmatrix}$

$1)~E_1\cdot A=B ~~~ \Rightarrow ~~~ E_1=B\cdot A^{-1} ~~~ \Rightarrow~~~ E_1=\begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}$

$2)~E_1\cdot B=A ~~~ \Rightarrow ~~~ E_1=A \cdot B^{-1} ~~~ \Rightarrow~~~ E_1=\begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}$

$3)~E_2\cdot A=C ~~~ \Rightarrow ~~~ E_2=C\cdot A^{-1} ~~~ \Rightarrow~~~ E_2=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -2 & 1 \end{pmatrix}$

$4)~E_3\cdot C=A ~~~ \Rightarrow ~~~ E_3=A \cdot C^{-1} ~~~ \Rightarrow~~~ E_3=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 2 & 1 \end{pmatrix}$

Here you can find inverse matrices and multiply matrices with a help of online calculators.