Let the three digit numbers A28, 3B9, and 62C, where A, B, and C are integers between 0 and 9, be di

Q: Wait, why are we allowed to perform $R_2 \to R_2 + 10R_3 + 100R_1$? Doesn't that change the determinant?

That is a fantastic question! In linear algebra, the property of determinants states that adding a scalar multiple of one or more rows to another row does not change the value of the determinant. Since we are adding $100$ times the first row and $10$ times the third row to the second row, the determinant remains identical to the original.

Q: How did you know to pick $100$ and $10$ as the multipliers?

I looked at the structure of the numbers! A three-digit number $xyz$ is $100x + 10y + z$. By looking at the columns of the determinant, I saw that the coefficients $100, 10,$ and $1$ were exactly what I needed to reconstruct the numbers $A28, 3B9,$ and $62C$ within the matrix itself.

Q: Can I just expand the determinant directly? It seems like a lot of work, but is it wrong?

Expanding it directly is not wrong, but it is a trap! You would end up with a massive algebraic expression involving $A, B,$ and $C$. While you might eventually prove it, the row operation method is the 'elegant' path that reveals the underlying symmetry of the problem.

Q: What if $k$ is zero? Does the divisibility rule still hold?

In the context of JEE problems, $k$ is typically a positive integer. If $k$ were zero, the condition 'divisible by $k$' would be undefined. We assume $k \in \mathbb{Z}^+$, which keeps our logic sound and our math clean.

Q: Does the order of the rows matter when I perform the operation?

The order of the rows is fixed by the determinant's definition, but the operation $R_2 \to R_2 + 10R_3 + 100R_1$ is specific to the positions of the digits in your numbers. If you swapped rows, you would have to adjust your multipliers accordingly to maintain the place-value structure.

Question

Let the three digit numbers $A28, 3B9$, and $62C$, where $A, B$, and $C$ are integers between $0$ and $9$, be divisible by a fixed integer $k$. Show that the determinant $\begin{vmatrix} A & 3 & 6 \ 8 & 9 & C \ 2 & B & 2 \end{vmatrix}$ is divisible by $k$.

Sigma Percentile · Accepted Answer

### The Hidden Symmetry of Numbers

Welcome, fellow traveler of the mathematical landscape. Today, we are not just solving a determinant problem; we are uncovering a hidden structure.

When you first look at the determinant

$$\Delta = \begin{vmatrix} A & 3 & 6 \ 8 & 9 & C \ 2 & B & 2 \end{vmatrix}$$

it might seem like a chaotic collection of variables and constants. But in the world of JEE Advanced, chaos is often just order in disguise. Our goal is to prove that this expression is divisible by a fixed integer $k$, given that $A28$, $3B9$, and $62C$ are all multiples of $k$.

### The Power of Place Value

Before we touch the determinant, let us pause and appreciate the beauty of our decimal system. Any three-digit number, say $xyz$, is simply a shorthand for $100x + 10y + z$.

When the problem tells us that $A28$, $3B9$, and $62C$ are divisible by $k$, it is whispering a secret to us. It is telling us that:

$$A28 = 100A + 10(2) + 8 = mk_1$$

$$3B9 = 100(3) + 10B + 9 = mk_2$$

$$62C = 100(6) + 10(2) + C = mk_3$$

Do you see the pattern? The coefficients $100$, $10$, and $1$ are the keys to the kingdom. If we can force our determinant to display these values, the proof will collapse into our hands like a house of cards.

### The Art of Row Manipulation

Now, let us look at our determinant again. We have rows of numbers, but they are currently 'disorganized' relative to our goal. We need to perform a transformation.

In linear algebra, we have the freedom to perform row operations without altering the value of the determinant. Specifically, the operation $R_i 	o R_i + mR_j + nR_k$ is our most powerful tool.

Imagine you are standing before this matrix. You want to transform the second row, $R_2$, into something meaningful. You look at the first row, $R_1$, and the third row, $R_3$. If you multiply $R_1$ by $100$ and $R_3$ by $10$, and then add them to $R_2$, what happens?

Let us calculate the new second row, which we shall call $R_2'$:

For the first column: $8 + 10(2) + 100(A) = 100A + 20 + 8 = A28$.

For the second column: $9 + 10(B) + 100(3) = 300 + 10B + 9 = 3B9$.

For the third column: $C + 10(2) + 100(6) = 600 + 20 + C = 62C$.

### The Grand Reveal

Look at what we have created! Our new determinant is:

$$\Delta = \begin{vmatrix} A & 3 & 6 \ A28 & 3B9 & 62C \ 2 & B & 2 \end{vmatrix}$$

Because we know that $A28$, $3B9$, and $62C$ are all divisible by $k$, we can factor $k$ out of the entire second row. This is the moment of triumph.

By the properties of determinants, if every element in a row shares a common factor, that factor can be pulled out in front of the determinant sign:

$$\Delta = k \cdot \begin{vmatrix} A & 3 & 6 \ \frac{A28}{k} & \frac{3B9}{k} & \frac{62C}{k} \ \ 2 & B & 2 \end{vmatrix}$$

Since the determinant is now expressed as $k$ multiplied by some value (which is an integer, given that $A, B, C$ are integers), it is mathematically certain that the original determinant $\Delta$ is **divisible by $k$**.

### Final Reflections

I want you to take a moment to appreciate what just happened. We didn't brute-force the calculation. We didn't get lost in a sea of algebraic expansion.

Instead, we used the properties of the system to reveal its inner logic. This is the essence of JEE Advanced mathematics—finding the elegant path through the complexity. You have the tools, you have the intuition, and now, you have the proof.

Animated Solution for Mathematics - Matrices and Determinants: Let the three digit numbers $A 28, 3 B 9$ , and $62 C$ , where $A, B$ , and $C$ are integers between $0$ and $9$ , be divisible by a fixed integer $k$ . Show that the determinant $A 82 39 B 6 C 2$ is divisible by $k$ .

Visualized Solution

Problem Statement

Decimal Expansion of Numbers

Analyzing the Determinant Structure

The Row Operation Strategy

Applying Operation to Column 1

Applying Operation to Column 2

Applying Operation to Column 3

The Transformed Determinant

Conclusion: Divisibility by $k$

The Sigma Insight: Properties of Determinants

The Hidden Symmetry of Numbers

The Power of Place Value

The Art of Row Manipulation

The Grand Reveal

Final Reflections

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