Question

Evaluate \(\begin{vmatrix}b-c&b&c\\a&c-a&c\\a&b&a-b\end{vmatrix}\).

(a) 2abc

(b) 2a{(b-c)(c-a+b)}

(c) 2b{(a-c)(a+b+c)}

(d) 2c{(b-c)(a-c+b)}

I got this question during an internship interview.

Question is taken from Properties of Determinants in section Determinants of Mathematics – Class 12

Answer 1

The correct choice is (b) 2a{(b-c)(c-a+b)}

For explanation: Δ=\(\begin{vmatrix}b-c&b&c\\a&c-a&c\\a&b&a-b\end{vmatrix}\)

Applying C2→C2-C3

Δ=\(\begin{vmatrix}b-c&b-c&c\\a&-a&c\\a&-a&a-b\end{vmatrix}\)

Applying C1→C1-C2

Δ=\(\begin{vmatrix}0&b-c&c\\2a&-a&c\\2a&-a&a-b\end{vmatrix}\)

Applying R2→R2-R3

Δ=\(\begin{vmatrix}0&b-c&c\\0&0&c-a+b\\2a&-a&a-b\end{vmatrix}\)

Expanding along C1, we get

Δ=2a{(b-c)(c-a+b)}