Question

What is the value of \(\begin{vmatrix}-bc & ca + ab & ca + ab \\ab + bc & -ca & ab + bc \\bc + ca & bc + ca & -ab \end {vmatrix}\) ?

(a) Σab

(b) (Σab)^2

(c) (Σab)^3

(d) (Σab)^4

This question was posed to me in an interview for job.

This key question is from Application of Determinants in portion Determinants of Mathematics – Class 12

Answer 1

Right option is (c) (Σab)^3

For explanation: Applying R1 –> R1 + R2 + R3

\(\begin{vmatrix}\Sigma ab & \Sigma ab & \Sigma ab \\ab + bc & -ca & ab + bc \\bc + ca & bc + ca & -ab \end {vmatrix}\)

This is equal to,

= Σab \(\begin{vmatrix}1 & 1 & 1 \\ab + bc & -ca & ab + bc \\bc + ca & bc + ca & -ab \end {vmatrix}\)

Applying C1 –> C1 – C2 and C2 –> C2 – C3

= Σab \(\begin{vmatrix}0 & 0 & 1 \\ \Sigma ab & -\Sigma ab & ab + bc \\0 & \Sigma ca & -ab \end {vmatrix}\)

= (Σab)^3