If {vmatrix} x & y & 0 \\ 0 & x & y \\ y & 0 & x {vmatrix} =0, then which one of the following is correct?

Correct Answer is : xyx/yyx is one of the cube roots of -1

x/y is one of the cube roots of unity

x is one of the cube roots of unity

y is one of the cube roots of unity

Correct Answer is : xyx/yyx is one of the cube roots of -1

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UPSC NDA I 2017 Mathematics Solved Paper

Question : 37 of 120

Marks: +1, -0

\begin{vmatrix} x & y & 0 \\ 0 & x & y \\ y & 0 & x \end{vmatrix} =0

, then which one of the following is correct?

Solution:

Concept:

The determinant of a

3 \times 3

matrix gives an equation linking

x

and

y

Explanation:

Given

\begin{vmatrix} x & y & 0 \\ 0 & x & y \\ y & 0 & x \end{vmatrix} = 0

Expand along the first row:

x \cdot (x \cdot x - y \cdot 0) - y \cdot (0 \cdot x - y \cdot y) + 0 = 0

Simplify:

x(x^2) - y( - y^2 ) = 0

, so

x^3 + y^3 = 0

Thus

x^3 = -y^3

Assuming

y \neq 0

, divide both sides by

y^3

\frac{x^3}{y^3} = -1

Take cube root:

\frac{x}{y} = \sqrt[3]{-1}

, which are the cube roots of

-1

Therefore

\frac{x}{y}

is one of the cube roots of

-1

Answer:

\frac{x}{y}

is one of the cube roots of

-1

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