The determinant $\begin{vmatrix} b^2 - ab & b - c & bc - ac \\ ab - a^2 & a - b & b^2 - ab \\ bc - ac & c - a & ab - a^2 \end{vmatrix}$ is equal to

None of these

The correct answer is Option (4) → None of these ##

We have, $\begin{vmatrix} b^2 - ab & b - c & bc - ac \\ ab - a^2 & a - b & b^2 - ab \\ bc - ac & c - a & ab - a^2 \end{vmatrix}$

On expanding along $R_1$, we get

$(b^2 - ab) [(a-b)(ab-a^2) - (c-a)(b^2-ab)] - (b-c) [(ab-a^2)^2 - (bc-ac)(b^2-ab)] + (bc-ac) [(ab-a^2)(c-a) - (a-b)(bc-ac)]$

$= (b^2 - ab) [(a^2b - a^3 - ab^2 + a^2b) - (cb^2 - abc - ab^2 + a^2b)] - (b-c) [a^2b^2 + a^4 - 2ab \cdot a^2 - (b^3c - ab^2c - ab^2c + a^2bc)] + (bc-ac) [(abc - a^2c - a^2b + a^3) - (abc - a^2c - b^2c + abc)]$

$= (b^2 - ab) [a^2b - a^3 - ab^2 + a^2b - cb^2 + abc + ab^2 - a^2b] - (b-c) [a^2b^2 + a^4 - 2a^3b - b^3c + ab^2c + ab^2c - a^2bc] + (bc-ac) [abc - a^2c - a^2b + a^3 - abc + a^2c + b^2c - abc]$

$= (b^2 - ab) [-a^3 - cb^2 + abc + a^2b] - (b-c) [a^2b^2 + a^4 - 2a^3b - b^3c + 2ab^2c - a^2bc] + (bc-ac) [-abc - a^2b + a^3 + b^2c]$

$= -a^3b^2 - cb^4 + ab^3c + a^2b^3 + a^4b + acb^3 - a^2b^2c - a^4b^2 - a^2b^3 - a^4b + 2a^3b^2 + b^4c - 2ab^3c + a^2b^2c + a^2b^2c + a^4c - 2a^3bc - b^3c^2 + 2ab^2c^2 - a^2bc^2 - ab^2c^2 - a^2cb^2 + a^3bc + b^3c^2 + a^2bc^2 + a^3bc - a^4c - ab^2c^2 = 0$