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Let A be an mxxn matrix. If there exists...

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. 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)]`

Text Solution

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The correct Answer is:
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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