Method 1:

Start with the matrix:


Augment it on the right with the identity matrix, ,
like this


 
Now we do row operations to get the identity on the left:

To get a zero where the -5 is,

Add -5 times row 1 to 3 times row 2.  It's a good idea
to write the number you're multiplying row by, on
the far left:



The result is:



Next we get a zero where the 1 is in row 1, column 2
by adding 1 times row 2 to -10 times row 1.



The result is:


 
Get a 1 where the 30 is by multiplying row 1
by 

Get a 1 where the 10 is by multiplying row 2
by 



The result is:


 
So the inverse of  or  is the part on the right:



That method works for all matrices.

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Here's a method that works only for 2x2 matrices:

Method 2:

Start with the matrix:



Find the determinant of the matrix:



Swap the upper left and lower right elements:



Change the signs of the upper right and lower left elements:



Divide every element by the -10 which was the determinant of the
original matrix:



Simplify the fractions:



Edwin