Find the value of the $\left|\begin{array}{cc}{x}^2-x+1& x-1\\ x+1& x+1\end{array}\right|$ at $x=1$.

$\left|\begin{array}{ccc}a+x& y& z\\ x& a+y& z\\ x& y& z+z\end{array}\right|$$=a^2\left(k\right)$. Find $k$.

$\left|\begin{array}{ccc}0& x{y}^2& x{z}^2\\ x^{2}y& 0& y{z}^2\\ x^{2}z& z{y}^2& 0\end{array}\right|$$=k{x}^3{y}^3{z}^3$, $k=?$

Using the properties of determinants, evaluate:

$\left|\begin{array}{ccc}3x& -x+y& -x+z\\ x-y& 3y& z-y\\ x-z& y-z& 3z\end{array}\right|$

Using the properties of determinants, evaluate:

$\left|\begin{array}{ccc}x+4& x& x\\ x& x+4& x\\ x& x& x+4\end{array}\right|$

If $\left|\begin{array}{ccc}a-b-c& 2a& 2a\\ 2b& b-c-a& 2b\\ 2c& 2c& c-a-b\end{array}\right|$$=(a+b+c{)}^k$. Find $k$.

Using the properties of determinants, prove that:

$\left|\begin{array}{lll}{y}^2{z}^2& yz& y+z\\ z^2{x}^2& zx& z+x\\ x^2{y}^2& xy& x+y\end{array}\right|=0$

Using the properties of determinants, prove that:

$\left|\begin{array}{ccc}y+z& z& y\\ z& z+x& x\\ y& x& x+y\end{array}\right|=4xyz$

Using the properties of determinants, prove that:

$\left|\begin{array}{ccc}{a}^2+2a& 2a+1& 1\\ 2a+1& a+2& 1\\ 3& 3& 1\end{array}\right|=(a-1{)}^3$