If `A=[[2,a,-3] , [0,2,5] , [1,1,3]]` then,find the value of a for which `A^(-1)` exists.

Options

A. `a=2`
B. `a!=2`
C. `a!=-2`
D. none of these

Answer

D

Explanation

We have
`A=|(2, lamda, -3),(0,2,5),(1,1,3)|`
Expanding along `R_(1)`
`|A|=2(6-5)-lamda(-5)-3(-2)=2+5lamda+6`
We know that `A^(-1)` exists if `A` is non singular matrix i.e. `|A|!=0`
`:.2+5lamda+6!=0`
`implies 5lamda!=-8`
`:. lamda!=(-8)/5`
So, `A^(-1)` exists if and only if `lamda!=(-8)/5`.