For how many values of 'x' in the closed interval `[-4,-1]` is the matrix `[(3,-1+x,2),(3,-1,x+2),(x+3,-1,2)]` singular ? (A) `2` (B) `0` (C) `3` (D) `1`

Options

A. 2
B. 0
C. 3
D. 1

Answer

D

Explanation

Since, the matrix
`[(3,-1+x,2),(3,-1,x+2),(x+3,-1,2)]` is singular.
`therefore|(3,-1+x,2),(3,-1,x+2),(x+3,-1,2)|=0`
`implies3[-2+(x+2)]+(1-x)`
`[6-(x+2)(x+3)]`
`+2[-3+(x+3)]=0`
`implies3x-x^(2)-5x+x^(3)+5x^(2)+2x=0`
`impliesx^(3)+4x^(2)+0`
`impliesx^(2)(x+4)=0`
`impliesx=0` or `x=-4`
Hence, for one values of x in the closed interval [-4,-1] the given matrix is singular.