Question

Let $A=a_{ij}$ be a matrix of order 3, where $a_{ij}=\{\begin{array}{cc}x& \text{if}i=j,x\in R\\ 1& \text{if}|i-j|=1\\ 0& \text{otherwise}\end{array}$

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 $f\left(x\right)=$ det A, then the function f(x) has both the maxima and minima

Answer 1

Answer: (B), (D)

The correct options are
B A is a symmetric matrix
D Let $f\left(x\right)=$ det A, then the function f(x) has both the maxima and minima

$A=\left(\begin{array}{ccc}x& 1& 0\\ 1& x& 1\\ 0& 1& x\end{array}\right)$

$A^T=\left(\begin{array}{ccc}x& 1& 0\\ 1& x& 1\\ 0& 1& x\end{array}\right)$

$A=A^T$ Hence this is a symmetric matrix

For any value of $x$, $A$ can't be a diagonal matrix

$|A|=x^3-2x$

if $x=2$,$|A|=4$

$f\left(x\right)=x^3-2x$ has both maxima and minima