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Question
Using properties of determinants, prove that
`|[x+y,x,x],[5x+4y,4x,2x],[10x+8y,8x,3x]|=x^3`
Solution
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
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