find the real value of x for which the matrix `=[(x+1,3,5),(1,x+3,5),(1,3,x+5)]` is non-singular.

Explanation

Let `A=[(x+1,3,5),(1,x+3,5),(1,3,x+5)]`
`therefore|A|=[(x+1,3,5),(1,x+3,5),(1,3,x+5)]`
Applying `C_(1)rarr C_(1)+C_(2)+C_(3),` then
`therefore|A|=[(x+9,3,5),(x+9,x+3,5),(x+9,3,x+5)]`
Applying `R_(2)rarrR_(2)-R_(1) and R_(3)rarrR_(3)-R_(1),` then
`|A|=|(x+9,3,5),(0,x,0),(0,0,x)|=x^(2)(x+9)`
`therefore` A is non-singular.
`therefore" " |A|!=0 rArr x^(2)(x+9)!=0 `
`therefore" " x!=0,-9`
Hence,` x in R - {0,-9}.`