Evaluating Determinants and Complex Numbers

The video tutorial explores the complex number omega, defined as cosine of 2π/3 plus i sine of 2π/3, and its role in solving a determinant problem. The instructor evaluates a 3x3 determinant, simplifies the expression, and applies Euler's formula to find the number of distinct complex numbers z that satisfy the equation. The solution reveals that z equals 0 is the only solution, highlighting the use of complex numbers and algebraic manipulation in problem-solving.

What is the complex number omega defined as in the problem?

• cosine of pi over 3 plus i sine of pi over 3

• cosine of 2 pi over 3 plus i sine of 2 pi over 3

• cosine of pi over 2 plus i sine of pi over 2

• cosine of 3 pi over 2 plus i sine of 3 pi over 2

What is the first step in evaluating the determinant?

• Multiply the entire determinant by omega

• Evaluate the sub 2 by 2 matrix

• Add all elements of the determinant

• Subtract omega from each element

What pattern is used when evaluating determinants?

• Checkerboard pattern

• Diagonal pattern

• Circular pattern

• Linear pattern

Which terms cancel out when simplifying the z terms?

• z omega squared and z omega

• z omega squared and negative z omega squared

• z squared and omega squared

• z and omega

How is omega expressed using Euler's formula?

• e to the 2 pi i

• e to the 2 pi over 3 i

• e to the i pi

• e to the 3 pi over 2 i

What is the value of omega to the third power?

• Square root of 3 over 2

• 0

• 1

• Negative 1

What is the equivalent angle for omega to the fourth power?

• 6 pi over 3

• 8 pi over 3

• 4 pi over 3

• 2 pi over 3

What is the final simplified form of the determinant equation?

• z plus omega equals 0

• z squared equals 0

• z minus omega equals 0

• z to the third power equals 0

What is the only solution to the equation z to the third power equals 0?

• z equals 1

• z equals omega

• z equals 0

• z equals negative 1

How many distinct complex numbers satisfy the determinant equation?

• 2

• 1

• 3

• 0