Say I have an elementary matrix associated with a row operation performed when doing Jordan Gaussian elimination so for example if I took the matrix that added 3 times the 1st row and added it to the 3rd row then the matrix would be the $3\times3$ identity matrix with a $3$ in the first column 3rd row instead of a zero.
Is there a way to quickly determine it's inverse (as in just by looking at it pretty much and without calculating cofactor matrix and transposing it.)
Thanks.
$\endgroup$2 Answers
$\begingroup$The inverses of elementary matrices are described in the properties section of the wikipedia page
$\endgroup$ $\begingroup$Yes, there is. If we show the matrix that adds line $j$ multiplied by a number $\alpha_{ij}$ to line $i$ by $E_{ij}$, then its inverse is simply calculated by $E^{-1} = 2I - E_{ij}$.
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