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.
For which of the following matrices, the number of left inverses is greater than the number of right inverses ?

Options

A. `[(1,2,4),(-3,2,1)]`
B. `[(3,2,1),(3,2,1)]`
C. `[(1,4),(2,-3),(2,-3)]`
D. `[(3,3),(1,1),(4,4)]`

Answer

C

Explanation

By observation there cannot be any left inverse for options (2) and (4). So we will check for options (1) and (3) only.
For option (1), let the left inverse be `[(a,b),(c,d),(0,0)]`. Then
`[(a,b),(c,d),(e,f)][(1,2,4),(-3,2,1)]=[(1,0,0),(0,1,0),(0,0,1)]`
Now, `a-3b=1, 2a+2b=0` and `4a+b=0` which is not possible. For option (3),
`[(a,b,c),(d,e,f)][(1,4),(2,-3),(5,4)]=[(1,0),(0,1)]`
`implies a+2b+5c=1, 4a-3b+4c=0, d+2e+5f=0, 4d-3e+4f=1`
Therefore, there are infinite number of left inverses.
`[(1,4),(2,-3),(5,4)][(a,b,c),(d,e,f)]=[(1,0,0),(0,1,0),(0,0,1)]`
`implies a+4d=1, 2a-3d=0` and `5a+4d=0`
which is not possible. Therefore, there is no right inverse.