GATE Overflow Test Series | Linear Algebra | Test 1 | Question: 3

Question

The value of the determinant $\begin{vmatrix}
2\alpha & \alpha + \beta & \alpha + \gamma\\
\beta + \alpha & 2\beta & \beta + \gamma\\
\gamma + \alpha & \gamma + \beta & 2\gamma
\end{vmatrix}$ is ________

Answer 1

Given determinant $\begin{vmatrix}
\alpha + \alpha & \alpha + \beta &  \alpha  + \gamma  \\
 \beta + \alpha  & \beta + \beta & \beta +  \gamma   \\
 \gamma + \alpha & \gamma + \beta  & \gamma + \gamma \end{vmatrix} $

The above determinant can be expressed as the sum of $8$ determinants.

Each of the $8$ determinants has either two identical columns or identical rows.

$\therefore$ Each of the resulting determinant is zero.

$\textbf{Short Trick:}$ Put $\alpha = \beta = \gamma = 1.$

So, the correct answer is $0.$

Comments

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