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Question

Let A=aij be a matrix of order 3, where
aij=x ;if i=j, xR1 ;if |ij|=10 ;otherwise ,
then which of the following hold(s) good:

A
for x=2, A is a diagonal matrix
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B
A is a symmetric matrix
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C
Let f(x)=detA, then the functionf(x) has only maxima
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D
Let f(x)=detA, then the functionf(x) has both the maxima and minima
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Solution

The correct options are
B A is a symmetric matrix
D Let f(x)=detA, then the functionf(x) has both the maxima and minima
aij=x ;if i=j, xR1 ;if |ij|=10 ;otherwise

A=x101x101x
|A|=x32x

If f(x)=x32x
f(x)=3x22
=(3x2)(3x+2)

f′′(x)=6x

So, x=23 is a point of minima and x=23 is a point of maxima.

Hence, f(x) has both the maxima and minima

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