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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 exi

Monjur Alahi

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.
Which of the following matrices is NOT left inverse of matrix `[(1,-1),(1,1),(2,3)]?`
A. `[(1/2,1/2,0),(-1/2,1/2,0)]`
B. `[(2,-7,3),(-1/2,1/2,0)]`
C. `[(-1/2,1/2,0),(-1/2,1/2,0)]`
D. `[(0,3,-1),(-1/2,1/2,0)]`


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Salim Ahmed Kawsar

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Correct Answer - C
As second row of all the options is same, we have to look at the elements of the first row. Let the left inverse be `[(a,b,c),(d,e,f)]`. Then
`[(a,b,c),(d,e,f)][(1,-1),(1,1),(2,3)]=[(1,0),(0,1)]`
`:. a+b+2c=1`
`-a+b+3c=1`, i.e., `b=(1-5c)/2, a=(1+c)/2`
Thus, matrices in the options (1), (2) and (4) are the inverses and matrix in option (3) is not the left inverse.

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