Let A=aij be a matrix of order 3, where aij=⎧⎪⎨⎪⎩x;ifi=j,x∈R1;if|i−j|=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 BA is a symmetric matrix D Let f(x)=detA, then the functionf(x) has both the maxima and minima aij=⎧⎪⎨⎪⎩x;ifi=j,x∈R1;if|i−j|=10;otherwise
⇒A=⎡⎢⎣x101x101x⎤⎥⎦ ⇒|A|=x3−2x
If f(x)=x3−2x ⇒f′(x)=3x2−2 =(√3x−√2)(√3x+√2)
⇒f′′(x)=6x
So, x=√2√3 is a point of minima and x=−√2√3 is a point of maxima.