problem regarding determinant

[nisar_cck]

find the value of determinant by using properties.write steps.

1st row x^3 y^3 z^3

2nd row x^2 y^2 z^2

3rd row y+z x+z x+y

we have to prove its value equal to

-( x^2 - y^2 )( y^2 - z^2 ) (z^2 - x^2 )

 

[stapel]

Without values for x, y, and z, I don't see any way of obtaining the "value" of the given determinant. All you can do is find an expression for it. Sorry.

Eliz.

 

[soroban]

Hello, nisar_cck!

Find the value of determinant by using properties.

. . . .x³ . . .y³ . . .z³

. . . .x² . . .y² . . .z²

. . .y+z . .x+z . .x+y

We have to prove its value equal to: .- (x² - y²)(y² - z²) (z² - x²)
.

I have no idea what "properties" we're supposed to use . . .

Did you expand the determinant?

. . x³·[y²(x+y) - z²(x+z)] - y³·[x²(x+y) - z²(y+z)] + z³·[x²(x+z) - y²(y+z)]

Evidently, this simplifies to that product.

 

[soroban]

Hello again, nisar_cck!

I expanded this monster:

. . x³·[y²(x+y) - z²(x+z)] - y³·[x²(x+y) - z²(y+z)] + z³·[x²(x+z) - y²(y+z)]

and many of the terms dropped out, leaving us with:

. . . . x⁴y² - x⁴z² - x²y⁴ + x²z⁴ + y⁴z² - y²z⁴

. . = .x⁴(y² - z²) - x²(y⁴ - z⁴) + y²z²(y² - z²)

. . = .x⁴(y² - z²) - x²(y² - z²)(y² + z²) + y²z²(y² - z²)

. . = .(y² - z²) [x⁴ - x²(y² + z²) + y²z²]

. . = .(y² - z²) (x⁴ - x²y² - x²z² + y²z²)

. . = .(y² - z²) (x⁴ - x²z² - x²y² + y²z²)

. . = .(y² - z²) [x²(x² - z²) - y²(x² - z²)]

. . = .(y² - z²) (x² - z²) (x² - y²)

. . = .- (x² - y²) (y² - z²) (z² - x²) . . . Whew!