Using properties of determinants, prove that $\left|\begin{array}{ccc}x+y & x & x \\ 5x+4y & 4x & 2x \\ 10x+8y & 8x & 3x\end{array}\right|=x^{3}.$

CBSE Class 12 · 2014 · DeterminantsLong Answer

Using properties of determinants, prove that

x + y 5 x + 4 y 10 x + 8 y x 4 x 8 x x 2 x 3 x = x^{3} .

Answer: Proved

Split column

1

as a sum (the

x

-part and the

y

-part) using the additivity of determinants in a column:

Δ = x 5 x 10 x x 4 x 8 x x 2 x 3 x + y 4 y 8 y x 4 x 8 x x 2 x 3 x .

In the second determinant, taking out common factors makes two columns proportional, so it is

0

. The first equals

x^{3} 1510148123 = x^{3} (1)

after reduction.

Δ = x^{3} .

Proved.

CBSE Class 12 Board · 2014 · Subjective