Let A be an `mxxn` matrix. If there exists a matrix L of type `nxxm` such that `LA=I_(n)`, then L is called left inverse of A. Similarly, if there exists a matrix R of type `nxxm` such that `AR=I_(m)`, then R is called right inverse of A.
For example, to find right inverse of matrix
`A=[(1,-1),(1,1),(2,3)]`, we take `R=[(x,y,x),(u,v,w)]`
and solve`AR=I_(3)`, i.e.,
`[(1,-1),(1,1),(2,3)][(x,y,z),(u,v,w)]=[(1,0,0),(0,1,0),(0,0,1)]`
`{:(implies,x-u=1,y-v=0,z-w=0),(,x+u=0,y+v=1,z+w=0),(,2x+3u=0,2y+3v=0,2z+3w=1):}`
As this system of equations is inconsistent, we say there is no right inverse for matrix A.
The number of right inverses for the matrix `[(1,-1,2),(2,-1,1)]` is

Options

A. 0
B. 1
C. 2
D. infinite

Answer

D

Explanation

Let right inverse be `[(a,b),(c,d),(e,f)]`. Then
`[(1,-1,2),(2,-1,1)][(a,b),(c,d),(e,f)]=[(1,0),(0,1)]`
Now, `a-c+2e=1`
`b-d+2f=0`
`2a-c+e=0`
`2b-d+f=1`
This system of equations has infinite solutions.