Question

Using the properties of determinants, prove the following:

`|[1,x,x+1],[2x,x(x-1),x(x+1)],[3x(1-x),x(x-1)(x-2),x(x+1)(x-1)]|=6x^2(1-x^2)`

Answer 1

`|[1,x,x+1],[2x,x(x-1),x(x+1)],[3x(1-x),x(x-1)(x-2),x(x+1)(x-1)]|`

`=x^2(1-x)|[1,x,x+1],[2,(x-1),(x+1)],[3,-(x-2),-(x+1)]| ` [Taking out x common from R₂ and x(1 − x) common form R₃]

`=x^2(1-x)(1+x)|[1,x,1],[2,(x-1),1],[3,-(x-2),-1]|` [Taking out (1 + x) common form C₃]

`=x^2(1-x^2)|[1,x,1],[1,-1,0],[4,2,0]| ` [Applying R₂→R₂−R₁ and R₃→R₃+R₁]

`=x^2(1-x^2)[1xx(2+4)-0+0] ` [Expanding along C₃]

`=6x^2(1-x^2)`