Question

Using properties of determinants, prove that

`|[x+y,x,x],[5x+4y,4x,2x],[10x+8y,8x,3x]|=x^3`

Answer 1

To prove:

`|[x+y,x,x],[5x+4y,4x,2x],[10x+8y,8x,3x]|=x^3`

Taking LHS, we get:

`|[x+y,x,x],[5x+4y,4x,2x],[10x+8y,8x,3x]|=|[x,x,x],[5x,4x,2x],[10x,8x,3x]|+|[y,x,x],[4y,4x,2x],[8y,8x,3x]|`

`=x^3|[1,1,1],[5,4,2],[10,8,3]|+x^2y|[1,1,1],[4,4,2],[8,8,3]|`

`=x^2|[0,0,1],[3,2,2],[7,5,3]|+0 ` (Using R₁ →R₁ - R₃ and R₂ →R₂ - R₃, in the first determinant)

`=x^3(15-14)=x^3` (2nd determinant is equal to zero as C₁ and C₂ are equal)

`=RHS`

Henced proved