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
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

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 matrixFor 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 minimaMathematics

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