If |6i -3i 1; 4 3i -1; 20 3 i| = x + iy, then (a) x = 3, y = 1 (b) x = 1, y = 3 (c) x = 0, y = 3 (d)

Q: Wait, why can't I just expand the determinant along the first row? It's a standard method, right?

You absolutely can, and you will get the correct answer eventually! However, in a high-stakes exam like JEE Advanced, time is your most precious resource. Expanding a $3 \times 3$ determinant with complex numbers involves multiple multiplications and additions, significantly increasing the probability of a sign error. Recognizing properties is the 'pro' way to solve it in seconds.

Q: How do I know what to factor out of a column?

Look at the target column you are comparing against. If you suspect $C_2$ is proportional to $C_3$, look at the ratio of their corresponding elements. Here, the first element of $C_2$ is $-3i$ and the first element of $C_3$ is $1$. The ratio is $-3i/1 = -3i$. If you factor out $-3i$ from $C_2$, you are essentially checking if the remaining elements match $C_3$.

Q: Is it always zero when columns are identical?

Yes, that is a fundamental property of determinants. Geometrically, a determinant represents the volume of a parallelepiped formed by the column vectors. If two columns are identical, the vectors are linearly dependent, meaning they lie in the same plane (or line), collapsing the volume to zero. It is a powerful tool to keep in your arsenal!

Question

If $\begin{vmatrix} 6i & -3i & 1 \ 4 & 3i & -1 \ 20 & 3 & i \end{vmatrix} = x + iy$, then

Options:

(A) $x = 3, y = 1$

(B) $x = 1, y = 3$

(C) $x = 0, y = 3$

(D) $x = 0, y = 0$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future engineers. Today, we are going to dismantle a problem that, at first glance, looks like a tedious exercise in complex number arithmetic.

We are given the determinant:

$$
\Delta = \begin{vmatrix} 6i & -3i & 1 \ 4 & 3i & -1 \ 20 & 3 & i \end{vmatrix} = x + iy
$$

Our mission is to find the values of $x$ and $y$.

### The Trap of Brute Force

If you were to expand this determinant directly using the first row, you would be dealing with terms like $(6i)(3i^2 - (-3)) - (-3i)(4i - (-20)) + (1)(12 - 60i)$.

While this is mathematically sound, it is a minefield of potential sign errors and algebraic slips. In an exam setting, we want to avoid this. We want to be elegant, fast, and utilize the **properties of determinants**.

### The Detective Work

Let us pause and inspect the columns. Specifically, look at the second column $C_2$ and the third column $C_3$.

The second column is $C_2 = [ -3i, 3i, 3 ]^T$. The third column is $C_3 = [ 1, -1, i ]^T$.

What if we factor out $-3i$ from the entire second column?

### The Algebraic Revelation

Let us perform the extraction. We pull $-3i$ out of the determinant, which is a valid property of determinants. Now, let us see what remains in $C_2$:

1. The first element: $\frac{-3i}{-3i} = 1$.
2. The second element: $\frac{3i}{-3i} = -1$.
3. The third element: $\frac{3}{-3i} = -\frac{1}{i}$.

We know that $\frac{1}{i} = -i$. Therefore, $-\frac{1}{i} = -(-i) = i$.

Suddenly, our second column has transformed into $[ 1, -1, i ]^T$. It is identical to the third column $C_3$. We have successfully shown that $C_2 = -3i 	imes C_3$.

### The Final Cancellation

This is the moment of truth. A fundamental property of determinants states that if any two columns are proportional, the determinant is zero.

Since we have established that $C_2$ and $C_3$ are linearly dependent, the value of the determinant is zero. We are left with the simple equation:

$$
0 = x + iy
$$

By equating the real and imaginary parts, we find that **$x = 0$** and **$y = 0$**.

### Conclusion

See how much time we saved? By looking for patterns rather than diving into the algebra, we turned a potentially messy calculation into a beautiful, one-step logical deduction.

This is the mindset of a topper. Always look for the property, always look for the shortcut, and always trust the elegance of mathematics.

Animated Solution for Mathematics - Matrices and Determinants: If $6 i 420 - 3 i 3 i 3 1 - 1 i = x + i y$ , then

Select Answer:

Visualized Solution

The Determinant Equation

Determinant Properties

Inspecting Columns $C_{2}$ and $C_{3}$

Elements of Column $C_{2}$

Factoring out $- 3 i$ from $C_{2}$

Modifying $C_{2}$ : First & Second Elements

Modifying $C_{2}$ : Third Element

Simplifying $- \frac{1}{i}$

Comparing the New $C_{2}$ with $C_{3}$

Applying the Determinant Property

Finding $x$ and $y$

The Sigma Insight: Properties of Determinants

Analyzing the Setup

The Trap of Brute Force

The Detective Work

The Algebraic Revelation

The Final Cancellation

Conclusion

Similar Questions

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Animated Solution for Mathematics - Matrices and Determinants: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ​6i420​−3i3i3​1−1i​​=x+iy, then

Select Answer:

Visualized Solution

The Determinant Equation

Determinant Properties

Inspecting Columns .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C2​ and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C3​

Elements of Column .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C2​

Factoring out .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } −3i from .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C2​

Modifying .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C2​: First & Second Elements

Modifying .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C2​: Third Element

Simplifying .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } −i1​

Comparing the New .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C2​ with .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C3​