Let `A=a_0` be a matrix of order 3, where `a_(i j)=x ; ifi=j ,x in R, 1 if|i-j|=1, 0;ot h e r w i s e` then when of the following Hold (s) good: for`x=2` , (a) `A` is a diagonal matrix (b) `A` is a symmetric matrix for`x=2` , (c) det `A` has the value equal to 6 (d) Let `f(x)=` , det `A ,` then the function `f(x)` has both the maxima and minima.

Options

A. for x = 2, A is a diagonal matrix
B. A is a symmetric matrix
C. for x = 2, det A has the value equal to 6
D. Let `f(x)=` det A, then the function f(x) has both the maxima and minima

Answer

B, D

Explanation

`a_("ij")={(x,,,"if "i=j","x in R),(1,,,"if "|i-j|=1),(0,,,"otherwise"):}`
`implies A=[(x,1,0),(1,x,1),(0,1,x)]`
`implies |A|=x^(3)-2x`
If `f(x)=x^(3)-2x`
`implies f'(x)=3x^(2)-2=(sqrt(3)x-sqrt(2)) (sqrt(3) x+sqrt(2))`
So `x=sqrt(2)/sqrt(3)` point of minima and `x= (-sqrt(2))/sqrt(3)` is maxima.