Assignment

If $a, b, c$ are three real numbers, then number of real roots of $\left|\begin{array}{ccc}x& c& -b\\ -c& x& a\\ b& -a& x\end{array}\right|=0$ is

If $p{\lambda }^4+q{\lambda }^3+r{\lambda }^2+s\lambda +t=\left|\begin{array}{ccc}{\lambda }^2+3\lambda & \lambda -1& \lambda +3\\ \lambda +1& 2-\lambda & \lambda -4\\ \lambda -3& \lambda +4& 3\lambda \end{array}\right|$, the value of $t$ is

Let $\left|\begin{array}{ccc}6i& -3i& 1\\ 4& 3i& -1\\ 20& 3& i\end{array}\right|=x+iy$, then

$\left|\begin{array}{lll}bc& b{c}^{\text{'}}+b^{\text{'}}c& b^{\text{'}}{c}^{\text{'}}\\ ca& c{a}^{\text{'}}+c^{\text{'}}a& c^{\text{'}}{a}^{\text{'}}\\ ab& a{b}^{\text{'}}+a^{\text{'}}b& a^{\text{'}}{b}^{\text{'}}\end{array}\right|$ is equal to

If $A$ is a square matrix of order $3$ and $det\left(A\right)=8$ then$det\left(adj\left(adj\left(A^2\right)\right)\right)$ is equal to $2^{\lambda }$, then $\lambda$ is equal to

If $A=\left[\begin{array}{ccc}0& 0& -1\\ 0& -1& 0\\ -1& 0& 0\end{array}\right]$ then $\mathrm{tr}\left(adj{A}^{50}\right)+\mathrm{tr}\left(A^{51}\right)+3\mathrm{tr}\left(A^{-1}\right)+5\left(tr\left({\left(A^2\right)}^{-1}\right)\right)$ is equal to

$A$ and $B$ are two non-singular matrices such that $A^6=I$ and $A{B}^2=BA\left(B\ne I\right)$, then find the value of $K$ so that $B^K=I$ is

If $A$ is an idempotent matrix satisfying ${\left(I-\frac{2}{5}A\right)}^{-1}=I-\alpha A$, then $\alpha$ is equal to