Embibe Experts Solutions for Exercise 5: Assignment

The system of equations
$x-y\cos \theta +z\cos 2\theta =0$

$-x\cos \theta +y-z\cos \theta =0$

$x\cos 2\theta -y\cos \theta +z=0$
has non-trivial solution for $\theta$ equals

If $A$ and $B$ are invertible square matrices of the same order then which of the following is/are correct?

Let $A=a_{ij}$ be a matrix of order $3\times 3$ where

$$\begin{gathered} a_{ij}=\left\{\begin{array}{l}x, \mathrm{if} i=j,x\in R \\ 1, \mathrm{if} \left|i-j\right|=1\\ 0, \mathrm{otherwise}\end{array}\right. \end{gathered}$$ 

then which of the following statement(s) is/are true?

Let ${\left[A_k\right]}_{n\times n}$ be a square matrix of order $n\times n$, such that $a_{ij}=\left\{\begin{array}{l}0, i\ne j\\ \frac{1}{k+j},i=j\end{array}\right.$ and ${\left[B_k\right]}_{n\times n}$ is its inverse matrix, then which is/are true?

Which of the following is/are true?

Let $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\\ 0& -2& 4\end{array}\right]$ and $a{A}^{-1}=b{A}^2+cA+dI$ where $\gcd$ of $a, d$ is one then

Let $A\left(x\right)=\left|\begin{array}{ccc}5+{\sin }^{2}x& {\cos }^{2}x& 4\sin 2x\\ {\sin }^{2}x& 5+{\cos }^{2}x& 4\sin 2x\\ {\sin }^{2}x& {\cos }^{2}x& 5+4\sin 2x\end{array}\right|$ for all $x\in R$ then the minimum value of $\left|adj(A\left(x\right)\right|=\lambda$. Then $\frac{\lambda }{100}$ is equal to

If $A=\left[\begin{array}{lll}a& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right]$, then value of $\left|adj(adj(adjA\left)\right)\right|\left|adj\left(A^{-1}\right)\right|=a^{\lambda }$ then $\lambda$ is equal to