Let `A=a_("ij")` be a matrix of order 3, where `a_("ij")={(x,,"if "i = j","x in R,),(1,,"if "|i-j|=1,,", then which of the following"),(0,,"otherwise",):}`
hold (s) good :

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.