Determinant $\left| \begin{matrix} b^{2} - ab & b - c & bc - ac \ ab - a^{2} & a - b & b^{2} - ab \ bc - ac & c - a & ab - a^{2} \end{matrix} ight|$ is equal to

Question

Determinant $\left| \begin{matrix} b^{2} - ab & b - c & bc - ac \ ab - a^{2} & a - b & b^{2} - ab \ bc - ac & c - a & ab - a^{2} \end{matrix} ight|$ is equal to

Dr. Abubacker · Accepted Answer

(c)
Let $\mathrm{\Delta} = \left| \begin{matrix}
b^{2} - ab & b - c & bc - ac \
ab - a^{2} & a - b & b^{2} - ab \
bc - ac & c - a & ab - a^{2}
\end{matrix} ight|$
$= (b - a)(b - a)\left| \begin{matrix}
b & b - c & c \
a & a - b & b \
c & c - a & a
\end{matrix} ight|$
$= (a - b)^{2}\left| \begin{matrix}
b & b & c \
a & a & b \
c & c & a
\end{matrix} ight|$ $(C_{2} ightarrow C_{2} + C_{3})$
$= 0$ $(\because$ two columns are same)

Q. Determinant $\left| \begin{matrix} b^{2} - ab & b - c & bc - ac \\ ab - a^{2} & a - b & b^{2} - ab \\ bc - ac & c - a & ab - a^{2} \end{matrix} \right|$ is equal to

Explanation

Reviewed by Dr. Abubacker

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