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Question

Let $$A= a_{ij}$$ be a matrix of order 3, where $$a_{ij}= \left\{\begin{matrix}x & \text{if}\ i = j, x \in R\\ 1 & \text{if}|i - j| = 1\\ 0 & \text{otherwise}\end{matrix}\right.$$
 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
for x=2, det A has the value equal to 6
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D
Let f(x)= det A, then the function f(x) has both the maxima and minima
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Solution

The correct options are
B A is a symmetric matrix
D Let $$f(x)= $$ det A, then the function f(x) has both the maxima and minima
$$A=\begin{pmatrix} x & 1 & 0 \\ 1 & x & 1 \\ 0 & 1 & x \end{pmatrix}$$
$$A^T=\begin{pmatrix} x & 1 & 0 \\ 1 & x & 1 \\ 0 & 1 & x \end{pmatrix}$$
$$A=A^T$$ Hence this is a symmetric matrix
For any value of $$x$$, $$A$$ can't be a diagonal matrix
$$|A|=x^3-2x$$
if $$x=2$$,$$|A|=4$$
$$f(x)=x^3-2x$$ has both maxima and minima

Mathematics

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