Let a, b, c be positive and not all equal. Show that the value of the determinant |a b c; b c a; c a

Q: Wait, why do we need the condition that $a, b, c$ are not all equal?

That is a fantastic observation! If $a=b=c$, then the sum of squares $(a-b)^2 + (b-c)^2 + (c-a)^2$ becomes zero, making the entire determinant zero. The problem specifically asks us to show the value is negative, and the condition 'not all equal' ensures that the sum of squares is strictly positive, which is essential for our proof.

Q: Why can't we just use energy conservation or some other physics principle?

I love the enthusiasm! However, this is a pure mathematical problem involving determinants and algebraic identities. While physics often uses these tools, here we must rely on the properties of determinants and algebraic factorization to reach the conclusion.

Q: Is there a faster way to solve this using properties of determinants?

Yes, you could use column operations like $C_1 \to C_1 + C_2 + C_3$ to factor out $(a+b+c)$ immediately. That is a very elegant approach! However, the expansion method we used is a fundamental technique that helps build a strong intuition for how these expressions behave.

Question

Let $a, b, c$ be positive and not all equal. Show that the value of the determinant $\begin{vmatrix} a & b & c \ b & c & a \ c & a & b \end{vmatrix}$ is negative.

Sigma Percentile · Accepted Answer

### Analyzing the Cyclic Symmetry

Imagine you are standing before a mathematical structure, a $3 	imes 3$ matrix, and you are asked to prove something about its determinant. In the world of JEE mathematics, these problems are invitations to see the hidden harmony of numbers.

Let us embark on a journey to decode the determinant:

$$\Delta = \begin{vmatrix} a & b & c \ b & c & a \ c & a & b \end{vmatrix}$$

### The Art of Expansion

Our first step is to peel back the layers. We expand this determinant along the first row to observe the terms as they emerge:

$$\Delta = a(bc - a^2) - b(b^2 - ac) + c(ab - c^2)$$

When we carefully distribute these terms, we simplify the expression:

$$\Delta = abc - a^3 - b^3 + abc + abc - c^3$$

Suddenly, the chaos settles into a beautiful, symmetric expression:

$$\Delta = 3abc - a^3 - b^3 - c^3$$

### The Hidden Identity

Now, look closely at that expression. It is the negative of the famous algebraic identity $a^3 + b^3 + c^3 - 3abc$. We can rewrite our determinant as:

$$\Delta = -(a^3 + b^3 + c^3 - 3abc)$$

We know the standard factorization for this identity is $x^3 + y^3 + z^3 - 3xyz = (x+y+z)(x^2+y^2+z^2-xy-yz-zx)$. By applying this, our expression transforms into:

$$\Delta = -(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$$

### The Master Stroke: Completing the Squares

We are almost there, but we need to be certain about the sign. To analyze the second bracket, we use a classic JEE manipulation by multiplying and dividing by $2$ to create perfect squares:

$$\Delta = -\frac{1}{2}(a+b+c)[2a^2+2b^2+2c^2-2ab-2bc-2ca]$$

By rearranging these terms, we reveal the sum of squares:

$$\Delta = -\frac{1}{2}(a+b+c)[(a-b)^2 + (b-c)^2 + (c-a)^2]$$

### The Final Revelation

Now, let us analyze the soul of this expression. We have a negative factor, a positive factor $(a+b+c)$, and a sum of squares.

Since $a, b, c$ are not all equal, at least one of the differences $(a-b)$, $(b-c)$, or $(c-a)$ must be non-zero. Therefore, the sum of squares is strictly positive.

A negative number multiplied by a positive number multiplied by another positive number is, inevitably, negative. We have arrived at the truth:

**$$\Delta < 0$$**

It is not just a calculation; it is a proof of the inherent nature of these numbers. Keep this logic in your toolkit, for it will serve you well in many more challenges to come.

Animated Solution for Mathematics - Matrices and Determinants: Let $a, b, c$ be positive and not all equal. Show that the value of the determinant $a b c b c a c a b$ is negative.

Visualized Solution

Defining the Determinant $Δ$

Expanding the Determinant

Simplifying the Expression

Grouping Terms

Applying Algebraic Identity

The Factored Form

The Sum of Squares Transformation

Completing the Squares

Sign Analysis: First Term

Sign Analysis: Sum of Squares

Final Conclusion

The Sigma Insight: Properties of Determinants

Analyzing the Cyclic Symmetry

The Art of Expansion

The Hidden Identity

The Master Stroke

The Final Revelation

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