Let A=begin{pmatrix}2 &1 6 & 4 end{pmatrix} a) Express A^{-1} as a product of elementary matrices b) Express A as a product of elementary matrices

Let A=begin{pmatrix}2 &1 6 & 4 end{pmatrix} a) Express A^{-1} as a product of elementary matrices b) Express A as a product of elementary matrices

Question
Matrices
asked 2020-11-17
Let \(A=\begin{pmatrix}2 &1 \\6 & 4 \end{pmatrix}\)
a) Express \(A^{-1}\) as a product of elementary matrices
b) Express A as a product of elementary matrices

Answers (1)

2020-11-18
Step 1
Given: \(A=\begin{pmatrix}2 &1 \\6 & 4 \end{pmatrix}\)
a) Express \(A^{-1}\) as a product of elementary matrices
b) Express A as a product of elementary matrices
Step 2
\(\begin{bmatrix}2 & 1 \\6 & 4 \end{bmatrix} \begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix} R_2 : R_2-3R_1 \begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix} R_2 : R_2-3R_1=\begin{bmatrix}1 & 0 \\-3 & 1 \end{bmatrix}E\)
\(=\begin{bmatrix}2 & 1 \\0 & 1 \end{bmatrix} \begin{bmatrix}1 & 0 \\-3 & 1 \end{bmatrix} R_2 : R_2 \div 2 = \begin{bmatrix}1/2 & 0 \\0 & 1 \end{bmatrix}E_2\)
\(=\begin{bmatrix}1 & 1/2 \\0 & 1 \end{bmatrix} \begin{bmatrix}1/2 & 0 \\-3 & 1 \end{bmatrix}R_1=R_1-\frac{R_2}{2}\begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix}R_1:R_1-\frac{R_2}{2}=\begin{bmatrix}1 & -1/2 \\ 0 & 1 \end{bmatrix}E_3\)
\(=\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix} \begin{bmatrix}2 & -1/2 \\-3 & 1 \end{bmatrix}\)
\(\therefore E_3 E_2 E_1 A =I\)
\(E_3 E_2 E_1I =A^{-1}\)
you can recheck it
\(A^{-1}=E_3E_2E_1=\begin{bmatrix}1 & -1/2 \\0 & 1 \end{bmatrix}\begin{bmatrix}1/2 & 0 \\0 & 1 \end{bmatrix}\begin{bmatrix}1 & 0 \\-3 & 1 \end{bmatrix}\)
\(=\begin{bmatrix}1 & -1/2 \\0 & 1 \end{bmatrix}\begin{bmatrix} 1/2 & 0 \\-3 & 1 \end{bmatrix} =\begin{bmatrix} 1/2 & 3/2 \\-3 & 1 \end{bmatrix}\)
\(=\begin{bmatrix} 2 & -1/2 \\-3 & 1 \end{bmatrix}\)
Step 3
\((A^{-1})^{-1}=(E_3E_2E_1)^{-1}\)
\(A=E_1^{-1}E_2^{-1}E_3^{-1}\)
\(E_1=\begin{bmatrix}1 & 0 \\-3 & 1 \end{bmatrix} , E_1^{-1}=\frac{1}{1}\begin{bmatrix}1 & 0 \\3 & 1 \end{bmatrix}\)
\(E_2=\begin{bmatrix}1/2 & 0 \\0 & 1 \end{bmatrix} , E_2^{-1}=\frac{1}{1/2}=\begin{bmatrix}1 & 0 \\0 & 1/2 \end{bmatrix}\)
\(E_3=\begin{bmatrix}1 & -1/2 \\0 & 1 \end{bmatrix} , E_3^{-1}=\frac{1}{1}=\begin{bmatrix}1 & 1/2 \\0 & 1 \end{bmatrix}\)
\(A=2\begin{bmatrix}1 & 0 \\3 & 1 \end{bmatrix}\begin{bmatrix}1 & 0 \\0 & 1/2 \end{bmatrix}\begin{bmatrix}1 & 1/2 \\0 & 1 \end{bmatrix}\)
\(=2\begin{bmatrix}1 & 0 \\3 & 1 \end{bmatrix}\begin{bmatrix}1 & 1/2 \\0 & 1/2 \end{bmatrix}=2\begin{bmatrix}1 & 1/2 \\3 & 3/2+1/2 \end{bmatrix}\)
\(=\begin{bmatrix}2 & 1 \\6 & 4 \end{bmatrix}\)
0

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