Question

Evaluate \(\begin{vmatrix}-a&b&c\\-2a+4x&2b-4y&2c+4z\\x&-y&z\end{vmatrix}\).

(a) 0

(b) abc

(c) 2abc

(d) -1

I had been asked this question by my college professor while I was bunking the class.

My doubt is from Properties of Determinants in chapter Determinants of Mathematics – Class 12

Answer 1

Correct answer is (a) 0

For explanation I would say: Δ=\(\begin{vmatrix}-a&b&c\\-2a+4x&2b-4y&2c+4z\\x&-y&z\end{vmatrix}\)

Using the properties of determinants, the given determinant can be expressed as a sum of two determinants.

Δ=\(\begin{vmatrix}-a&b&c\\-2a&2b&2c\\x&-y&z\end{vmatrix}\)+\(\begin{vmatrix}-a&b&c\\4x&-4y&4z\\x&-y&z\end{vmatrix}\)

Δ=2\(\begin{vmatrix}-a&b&c\\-a&b&c\\x&-y&z\end{vmatrix}\)+4\(\begin{vmatrix}-a&b&c\\x&-y&z\\x&-y&z\end{vmatrix}\)

Since two rows are similar in each of the determinants, the determinant is 0.