Question

Obtain the inverse of the matrix `A=[{:(2,3),(1,1):}]` using elementary operations.

Answer 1

Write `A=IA. i.e. [{:(,0,1,2),(,1,2,3),(,3,1,1):}]=[{:(,1,0,0),(,0,1,0),(,0,1,1):}]A`
or `[{:(,1,2,3),(,0,1,2),(,3,1,1):}]=[{:(,0,1,0),(,1,0,0),(,0,0,1):}]A ("applying "R_(1) Leftrightarrow R_(2)`
`or [{:(,1,2,3),(,0,1,2),(,0,-5,-8):}]=[{:(,0,1,0),(,1,0,0),(,0,-3,1):}]A ("applying "R_(3) rightarrow R_(3)-3R_(1))`
or `or [{:(,1,0,-1),(,0,1,2),(,0,-5,-8):}]=[{:(,-2,1,0),(,1,0,0),(,0,-3,1):}]A ("applying "R_(1) rightarrow R_(1)-2R_(2))`
`or [{:(,1,0,-1),(,0,1,2),(,0,0,2):}]=[{:(,-2,1,0),(,1,0,0),(,5,-3,1):}]A ("applying "R_(3) rightarrow R_(3)+5R_(2))`
`or [{:(,1,0,-1),(,0,1,2),(,0,0,1):}]=A[{:(,-2,1,0),(,1,0,0),(,(5)/(2),(-3)/(2),(1)/(2)):}]A ("applying "R_(3) rightarrow (1)/(2)R_(3))`
`or [{:(,1,0,0),(,0,1,2),(,0,0,1):}]=A[{:(,(1)/(2),(-1)/(2),(1)/(2)),(,1,0,0),(,(5)/(2),(-3)/(2),(1)/(2)):}]A ("applying "R_(1) rightarrow R_(1)+R_(3))`
`or [{:(,1,0,0),(,0,1,0),(,0,0,1):}]=A[{:(,(1)/(2),(-1)/(2),(1)/(2)),(,-4,3,-1),(,(5)/(2),(-3)/(2),(1)/(2)):}]A ("applying "R_(2) rightarrow R_(2)-2R_(3))`
`"Hence"=A^(-1)[{:(,(1)/(2),(-1)/(2),(1)/(2)),(,-4,3,-1),(,(5)/(2),(-3)/(2),(1)/(2)):}]`