Question

Evaluate \(\begin{vmatrix}1+m&n&q\\m&1+n&q\\n&m&1+q\end{vmatrix}\).

(a) -1(1+m+n+q)

(b) 1+m+n+q

(c) 1+2q

(d) 1+q

I have been asked this question during a job interview.

I want to ask this question from Properties of Determinants topic in division Determinants of Mathematics – Class 12

Answer 1

Right answer is (a) -1(1+m+n+q)

For explanation: Given that, Δ=\(\begin{vmatrix}1+m&n&q\\m&1+n&q\\n&m&1+q\end{vmatrix}\)

Applying C1→C1+C2+C3

Δ=\(\begin{vmatrix}1+m+n+q&n&q\\1+m+n+q&1+n&q\\1+m+n+q&m&1+q\end{vmatrix}\)=(1+m+n+q)\(\begin{vmatrix}1&n&q\\1&1+n&q\\1&m&1+q\end{vmatrix}\)

Applying R1→R2-R1

Δ=(1+m+n+q)\(\begin{vmatrix}0&1&0\\1&1+n&q\\1&m&1+q\end{vmatrix}\)

Expanding along the first row, we get

Δ=(1+m+n+q)(0-1(1+q-q)+0)

Δ=-1(1+m+n+q).