Question

The value of $\begin{vmatrix} a & a+b & a+2b \\ a+2b & a & a+b \\ a+b & a+2b & a \end{vmatrix}$ is equal to

• A. $9a^{2}(a+b)$
• B. $9b^{2}(a+b)$
• C. $a^{2}(a+b)$
• D. $b^{2}(a+b)$

Hint:

Adding all rows (or columns) is a powerful trick when each row has a common sum. Factor out the common element before expanding.

Answer 1

The Correct Option is B

Step 1: Conceptual Understanding:
Use row operations to simplify the determinant.
Step 2: Explanation in Detail:
Apply $R_1 \to R_1 + R_2 + R_3$. Each element of the new first row equals $3(a+b)$, so factor out $3(a+b)$. After further row and column operations the determinant evaluates to $9b^2(a+b)$.
Step 3: Therefore, Stating the Final Answer
The value of the determinant is $9b^2(a+b)$.