Animated Solution for Mathematics - Matrices and Determinants: If $a_1,a_2,a_3,\dots ,a_n,\dots$ are in G.P., then the value of the determinant $\left|\begin{array}{ccc}\log a_n& \log a_{n+1}& \log a_{n+2}\\ \log a_{n+3}& \log a_{n+4}& \log a_{n+5}\\ \log a_{n+6}& \log a_{n+7}& \log a_{n+8}\end{array}\right|$ is

Imagine you are staring at a $3\times 3$ determinant filled with logarithms of terms from a Geometric Progression (G.P.). At first glance, it looks like a wall of intimidating, complex expressions.

You see subscripts like $n+8$ and you might feel the urge to start expanding it immediately. But stop. Take a breath.

In the world of JEE Advanced, whenever you see a structure this rigid, there is almost always a hidden symmetry waiting to be uncovered. Today, we are going to dismantle this determinant not with brute force, but with the elegance of mathematical properties.

Secret

First, let us recall the soul of a Geometric Progression. If we have a sequence $a_1,a_2,a_3,\dots$, the $k$-th term ahead of $a_n$ is defined as $a_{n+k}=a_n{r}^k$, where $r$ is the common ratio.

Now, look at the determinant. Every single entry is a logarithm. Let us apply the logarithm to our general term:

Using the fundamental laws of logarithms, specifically the product rule $\log (xy)=\log (x)+\log (y)$ and the power rule $\log (x^k)=k\log (x)$, this expression transforms beautifully into:

Do you see what happened? We have converted a multiplicative relationship into an additive one. This is the 'Aha!' moment.

The sequence of logarithms is no longer a G.P.; it has transformed into an Arithmetic Progression (A.P.) with a first term of $\log (a_n)$ and a common difference of $\log (r)$.

Now, let us populate our determinant with this new knowledge. The first row consists of $\log (a_n)$, $\log (a_n)+\log (r)$, and $\log (a_n)+2\log (r)$.

The second row follows the same pattern, starting from $\log (a_{n+3})$, which is $\log (a_n)+3\log (r)$. If you write this out, you will see a grid where the $\log (a_n)$ term is present in every single cell.

This is the 'noise' we need to eliminate. In determinant theory, we have a powerful tool: column operations. These operations allow us to manipulate the columns without changing the value of the determinant.

Let us perform two strategic operations: 1. $C_2\to C_2-C_1$ 2. $C_3\to C_3-C_2$

Watch what happens when we subtract the first column from the second. For any row, the $\log (a_n)$ terms cancel out perfectly! We are left with the difference of the log terms, which is simply $\log (r)$.

For example, in the first row:

This happens for every row. Suddenly, the entire second column collapses into a vector of $\log (r)$.

If we repeat this for the third column by subtracting the second column, the same magic occurs. The $\log (a_n)$ terms vanish, and the difference between the multipliers of $\log (r)$ is again $1$. Thus, the third column also becomes a vector of $\log (r)$.

We are left with a determinant where the second and third columns are identical:

There is a golden rule in linear algebra: if any two columns of a determinant are identical, the determinant is zero. We do not need to expand this. We do not need to calculate a single complex product.

The structure itself dictates the answer. The answer is $0$.

This problem is a masterclass in why we study properties. It teaches us that when faced with complexity, we should look for the underlying pattern.

The G.P. was just a disguise for an A.P., and the determinant was just a stage for a beautiful cancellation. Keep this logic in your toolkit, and you will find that even the most intimidating problems have a simple, elegant heart.