Answer to Question #198157 in Linear Algebra for anu

Question #198157

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.


1
Expert's answer
2021-05-31T06:46:40-0400

A=(271341815)B=(815341271)C=(271341273)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) E1A=B      E1=BA1      E1=(001010100)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) E1B=A      E1=AB1      E1=(001010100)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) E2A=C      E2=CA1      E2=(100010021)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) E3C=A      E3=AC1      E3=(100010021)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.


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