Name: Solving the Systems Using Cramer's Rule
Uploaded: 2019-07-19
Duration: 15 min 2 s
Description: In this video, you will learn how to solve the system of linear equations with the help of cramer's rule? So let's watch this video to know more about this concept.

Question

If

|\begin{array}{l} x & x^{2} & x^{3} - 1 \\ y & y^{2} & y^{3} - 1 \\ z & z^{2} & z^{3} - 1 \end{array}| = 0

, prove that

x y z = 1

when

x, y, z

are non-zero and unequal.

Accepted Answer

Consider $∆ = |\begin{array}{l} x & x^{2} & x^{3} - 1 \\ y & y^{2} & y^{3} - 1 \\ z & z^{2} & z^{3} - 1 \end{array}|$

$\Rightarrow ∆ = |\begin{array}{l} x & x^{2} & x^{3} \\ y & y^{2} & y^{3} \\ z & z^{2} & z^{3} \end{array}| + |\begin{array}{l} x & x^{2} & - 1 \\ y & y^{2} & - 1 \\ z & z^{2} & - 1 \end{array}|$

$\Rightarrow ∆ = x y z |\begin{array}{l} 1 & x & x^{2} \\ 1 & y & y^{2} \\ 1 & z & z^{2} \end{array}| - |\begin{array}{l} x & x^{2} & 1 \\ y & y^{2} & 1 \\ z & z^{2} & 1 \end{array}|$

$x y z = 1$ $|\begin{array}{l} 1 & x^{2} & x \\ 1 & y^{2} & y \\ 1 & z^{2} & z \end{array}|$ $- (- 1)$ $|\begin{array}{l} 1 & x^{2} & x \\ 1 & y^{2} & y \\ 1 & z^{2} & z \end{array}|$ (Interchanging two rows or columns changes the sign of determinant)

$\Rightarrow ∆ = (- x y z + 1)$ $|\begin{array}{l} 1 & x^{2} & x \\ 1 & y^{2} & y \\ 1 & z^{2} & z \end{array}|$

$\Rightarrow ∆ = (- x y z + 1)$ $|\begin{array}{l} 1 & x^{2} & x \\ 0 & y^{2} - x^{2} & y - x \\ 0 & z^{2} - x^{2} & z - x \end{array}|$ (Applying $R_{2} \to R_{2} - R_{1}$ and $R_{3} \to R_{3} - R_{1}$ )

$\Rightarrow ∆ = (- x y z + 1) (y - x) (z - x)$ $|\begin{array}{l} 1 & x^{2} & x \\ 0 & y + x & 1 \\ 0 & z + x & 1 \end{array}|$

$\Rightarrow ∆ = (- x y z + 1) (y - x) (z - x) [1 (y + x - z - x) - 0 + 0]$

$\Rightarrow ∆ = (- x y z + 1) (y - x) (z - x) (y - z)$

Now, $∆ = 0 \Rightarrow (- x y z + 1) (y - x) (z - x) (y - z) = 0$

$\Rightarrow (- x y z + 1) = 0 or (y - x) = 0 or (z - x) = 0 or (y - z) = 0$

$\Rightarrow x y z = 1 or y = x or z = x or y = z$

But $x, y, z$ are not equal.

$\Rightarrow x y z = 1$

If $|\begin{array}{l} x & x^{2} & x^{3} - 1 \\ y & y^{2} & y^{3} - 1 \\ z & z^{2} & z^{3} - 1 \end{array}| = 0$ , prove that $x y z = 1$ when $x, y, z$ are non-zero and unequal.

Important Questions on Determinants

Learn and Prepare for any exam you want !

If xx2x3-1yy2y3-1zz2z3-1=0, prove that x y z=1 when x,y,z are non-zero and unequal.

Important Questions on Determinants

Learn and Prepare for any exam you want !

If $|\begin{array}{l} x & x^{2} & x^{3} - 1 \\ y & y^{2} & y^{3} - 1 \\ z & z^{2} & z^{3} - 1 \end{array}| = 0$ , prove that $x y z = 1$ when $x, y, z$ are non-zero and unequal.