Complex determinant example | Imaginary and complex numbers | Precalculus | Khan Academy

TL;DR

Evaluating a complex determinant leads to the conclusion that the only solution is z = 0.

Transcript

Let omega be the complex number cosine of 2 pi over 3 plus i sine of 2 pi over 3, then the number of distinct complex numbers z satisfying this determinant equaling 0. So we have this 3 by 3 determinant equaling 0. So let's just evaluate this determinant and see if we can solve for z, or figure out how many complex numbers z that we get satisfying ... Read More

Key Insights

• 🍉 Evaluating complex determinants requires breaking them down into sub-determinants and simplifying each term.
• 🎮 The video simplifies the determinant to a polynomial equation, z^3 = 0.
• #️⃣ The associated exponential notation for complex numbers, such as Euler's formula, can make calculations more manageable.

Questions & Answers

Q: How does the video simplify the complex determinant?

The video breaks down the determinant into sub-determinants and simplifies each term, eventually leading to a polynomial equation.

Q: What is the simplified polynomial equation?

The polynomial equation is z^3 = 0.

Q: What is the significance of the polynomial equation?

The only complex number that satisfies the equation is z = 0, which means there is only one distinct complex number that solves the equation.

Q: Why is z = 0 the only solution?

Any number raised to the power of 3 equals 0 is only true when the number itself is 0, which means z = 0 is the only solution.

Summary & Key Takeaways

• The video explains the process of evaluating a 3x3 determinant involving complex numbers.

• The determinant is simplified to a polynomial equation, which is further simplified to z^3 = 0.

• The only solution to the equation is z = 0.