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

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

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:
$$A^(-1)=E_1xxE_2$$
$$=[[1,0],[-1,1]]*[[1/2,0],[0,1]]$$
$$=[[1/2,0],[-1/2,1]]$$

### Relevant Questions

Use elementary matrices to write ? in row reduced echelon form. Use the elementary matrices to find the inverse of
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Which of the following matrices is elementary matrix?
a) $$\begin{bmatrix}0 & 3 \\1 & 0 \end{bmatrix}$$
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d) $$\begin{bmatrix}2 & 0 \\0 & 2 \end{bmatrix}$$
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a)Find elementary matrices $$E_1 \text{ and } E_2$$ such that $$C=E_2E_1A$$
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and suppose that we have the following row reduction to its PREF B
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