The value of $x,$ if $\left| \begin{matrix} -x & 1 & 0 \\ 1 & -x & 1 \\ 0 & 1 & -x \end{matrix} \right| = 0$ is equal to

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

If $x^2+y^2+z^2 \neq 0, \quad x=cy+bz, \quad y=az+cx$ and $z=bx+ay$, then $a^2+b^2+c^2+2abc$ is equal to

The set of all values of $t \in R$,for which the matrix $\left[\begin{array}{ccc}e^t & e^{-t}(\sin t-2 \cos t) & e^{-t}(-2 \sin t-\cos t) \\e^t & e^{-t}(2 \sin t+\cos t) & e^{-t}(\sin t-2 \cos t) \\e^t & e^{-t} \cos t & e^{-t} \sin t \end{array}\right]$ is invertible.

$\left|\begin{array}{ccc}x & 3x+2 & 2x-1 \\ 2x-1 & 4x & 3x+1 \\ 7x-2 & 17x+6 & 12x-1\end{array}\right|=0$ is true for

Evaluate $\left|\begin{array}{ccc}1 & x & y \\ 1 & x+y & y \\ 1 & x & x+y\end{array}\right|$