Augment the matrix with the identity matrix on the right.
Then do row operations to get the identity on the left,
and the inverse will be on the right:
-----------------------------------------------
Swap rows 1 and 2 to get a non-zero element
in the upper left corner:
-----------------------------------------------
We get column 1 so that all its elements are
0 except the element in row 1.
The least common multiple of the non-zero
elements in column 1 is 2, so we get all
the non-zero elements in column 1 to be 2
except row 1, and we get it to be -2
So we only need to
multiply row 1 by -2
Add row 1 to rows 3 and 4, and restore
row 1:
-----------------------------------------------
We get column 2 so that all its elements are
0 except the element in row 2.
The least common multiple of the non-zero
elements in column 2 is 1, so we get all
the non-zero elements in column 2 to be 1
except row 2, and we get it to be -1
So we only need to
multiply row 2 by -1
Add row 2 to rows 1 and 4, and restore
row 2:
-----------------------------------------------
We get column 3 so that all its elements are
0 except the element in row 3.
The least common multiple of the non-zero
elements in column 3 is 6, so we get all
the non-zero elements in column 3 to be 6
except row 3, and we get it to be -6
So we need to
multiply row 2 by 3
multiply row 3 by 3
multiply row 4 by -2
Add row 3 to rows 2 and 4, and
restore row 3
-----------------------------------------------
We get column 4 so that all its elements are
0 except the element in row 4.
The least common multiple of the non-zero
elements in column 4 is 3, so we get all
the non-zero elements in column 4 to be 3
except in row 4, and we get it to be -3
So we need to
multiply row 1 by 3
multiply row 2 by -1
multiply row 3 by -1
multiply row 4 by -3
Add row 4 to rows 1, 2, and 3, and restore row 4
Get 1's on the diagonal
Divide row 1 by 3
Divide row 2 by -3
Divide row 3 by 2
-------------------------------------
Now that we have the identity matrix on the left
of the petition, the inverse of the given matrix
is the 4×4 matrix on the right of the petition:
Edwin
