Find the inverse of the matrix using elementary matrices. [[2,0],[1,1]]

Find the inverse of the matrix using elementary matrices. [[2,0],[1,1]]

Question
Matrices
asked 2021-01-17
Find the inverse of the matrix using elementary matrices. \([[2,0],[1,1]]\)

Answers (1)

2021-01-18
Let the given matrix be,
\(A=[[2,0],[1,1]]\)
Now transform the given matrix in to unit matrix I by performing column reduction to get,
\(A=[[2,0],[1,1]]\)
\(=[[2,0],[1,1]](c_1 -> c_1 - c_2)\)
\(=[[1,0],[0,1]](c_1 -> 1/2 * c_1)\)
=I
Now, construct sequence of elementary matrices such that E2*E1 A=I
The column operation are,
\(E_1=[[1,0],[0,1]](c_1 -> c_1 - c_2)\)
\(=[[1,0],[-1,1]]\)
\(E_2=[[1,0],[0,1]](c_1 -> 1/2 * c_1)\)
\(=[[1/2,0],[0,1]]\)
Then the inverse of the matrix by elementary matrices is given by:
\(A^(-1)=E_1xxE_2\)
\(=[[1,0],[-1,1]]*[[1/2,0],[0,1]]\)
\(=[[1/2,0],[-1/2,1]]\)
0

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