Question

How do you find the inverse of `[(6,7), (-6,7)]`?

Answer 1

Append an identity matrix to the right of the given matrix
Perform elementary row operations until you obtain an identity matrix on the left.
The matrix on the right is the inverse.

Explanation:

Let #A= [ (6,7), (-6,7) ]#

Append an identity matrix on the right:

#A|I = [ (6,7,|,1,0), (-6,7,|,0,1) ]#

Perform elementary row operations

`R_1 + R_2 to R_2`

#[ (6,7,|,1,0), (0,7,|,1,1) ]#

`R_1 - R_2 to R_1`

#[ (6,0,|,0,-1), (0,7,|,1,1) ]#

`R_1/6`

#[ (1,0,|,0,-1/6), (0,7,|,1,1) ]#

`R_2/7`

#[ (1,0,|,0,-1/6), (0,1,|,1/7,1/7) ]#

There is an identity matrix on the left, therefore, the matrix on the right is the inverse:

#A^-1 = [ (0,-1/6), (1/7,1/7) ]#