For what values of k, the system of linear equations

x + y + z = 2
2x + y - z = 3
3x + 2y + kz = 4

has a unique solution?

Using properties of determinants, prove that

Using the properties of determinants, solve the following for x:

314.

If  find the value of x.

Use Properties of determinants, prove that:

If  then write the value of x. 

Using properties of determinants prove the following:

Using properties of determinants, prove that 

Use product to solve the system of equations x + 3z = 9,
–x + 2y – 2z = 4, 2x – 3y + 4z = –3

Using properties of determinants, prove that

$\left|\begin{array}{ccc}1& 1& 1+3\mathrm{x}\\ 1+ 3\mathrm{y}& 1& 1\\ 1& 1+3\mathrm{z}& 1\end{array}\right| = 9 (3\mathrm{xyz} + \mathrm{xy} + \mathrm{yz} + \mathrm{zx})$