For positive numbers x, y and z, the numerical value of the determinant |1 log_x y log_x z; log_y x

Q: Wait, why can't we just expand the determinant using the standard formula? It seems like a lot of work, but wouldn't it work?

Expanding a 3x3 determinant with logarithmic terms is a recipe for algebraic disaster. While theoretically possible, it would lead to a massive expression that is nearly impossible to simplify. In JEE Advanced, we look for structural elegance, not brute force. Always look for properties first!

Q: What if $x, y, z$ are equal to 1? The logarithm would be undefined.

Excellent observation! The problem specifies $x, y, z$ are positive numbers. In the context of logarithms, the base must be positive and not equal to 1. We assume $x, y, z \neq 1$ for the expression to be well-defined.

Q: Is there a shortcut to seeing the answer is zero?

Yes! Once you see the structure, you can recognize that the rows are proportional. If rows are proportional, the determinant is zero. This is a classic pattern in competitive exams.

Question

For positive numbers $x, y$ and $z$, the numerical value of the determinant $\begin{vmatrix} 1 & \log_x y & \log_x z \ \log_y x & 1 & \log_y z \ \log_z x & \log_z y & 1 \end{vmatrix}$ is .........

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are presented with a determinant:

$$\Delta = \begin{vmatrix} 1 & \log_x y & \log_x z \ \log_y x & 1 & \log_y z \ \log_z x & \log_z y & 1 \end{vmatrix}$$

At first glance, the different bases $x, y,$ and $z$ create a chaotic environment. However, in the world of JEE Advanced, such complexity is merely an opportunity to find underlying order.

### The Master Key: Change of Base

The primary obstacle is the inconsistency of the bases. We have base $x$ in the first row, base $y$ in the second, and base $z$ in the third.

We cannot perform row operations effectively when the bases are mismatched. This is where the **Change of Base Theorem** comes to our rescue:

$$\log_a b = \frac{\ln b}{\ln a}$$

By converting every term to a common base (the natural logarithm, $\ln$), we can transform the entire structure into a manageable form.

### The Transformation

Let us rewrite each row using the common base $\ln$. For the first row, we have $1 = \frac{\ln x}{\ln x}$, $\log_x y = \frac{\ln y}{\ln x}$, and $\log_x z = \frac{\ln z}{\ln x}$.

Notice the common denominator $\ln x$. We apply the same logic to the second row (denominator $\ln y$) and the third row (denominator $\ln z$). Our determinant now becomes:

$$\Delta = \begin{vmatrix} \frac{\ln x}{\ln x} & \frac{\ln y}{\ln x} & \frac{\ln z}{\ln x} \ \frac{\ln x}{\ln y} & \frac{\ln y}{\ln y} & \frac{\ln z}{\ln y} \ \frac{\ln x}{\ln z} & \frac{\ln y}{\ln z} & \frac{\ln z}{\ln z} \end{vmatrix}$$

### The Elegance of Factoring

Now, we observe the structure carefully. We can factor out $\frac{1}{\ln x}$ from the first row, $\frac{1}{\ln y}$ from the second row, and $\frac{1}{\ln z}$ from the third row.

This leaves us with the following expression:

$$\Delta = \frac{1}{\ln x \cdot \ln y \cdot \ln z} \begin{vmatrix} \ln x & \ln y & \ln z \ \ln x & \ln y & \ln z \ \ln x & \ln y & \ln z \end{vmatrix}$$

### The Final Revelation

Look at the resulting determinant. The rows are identical!

In linear algebra, a fundamental property states that if any two rows (or columns) of a determinant are identical, the value of the determinant is zero. Since all three rows are identical in this case, the determinant is undeniably zero.

Multiplying this zero by our external factors leaves us with a final answer of **$0$**. This problem teaches us that even the most complex-looking expressions often hide a simple, elegant truth.

Animated Solution for Mathematics - Matrices and Determinants: For positive numbers $x, y$ and $z$ , the numerical value of the determinant $1 lo g_{y} x lo g_{z} x lo g_{x} y 1 lo g_{z} y lo g_{x} z lo g_{y} z 1$ is .........

Enter Numerical Value:

Visualized Solution

The Logarithmic Determinant

Different Bases

Change of Base Theorem

Expanding Row 1

Expanding Row 2

Expanding Row 3

Common Factor in $R_{1}$

Common Factor in $R_{2}$

Common Factor in $R_{3}$

Identical Rows

Zero Determinant Property

Final Answer

The Sigma Insight: Properties of Determinants

Analyzing the Setup

The Master Key

The Transformation

The Elegance of Factoring

The Final Revelation

Similar Questions

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