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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)`
Solution
`|[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)`
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