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

Khan Academy

29 Dec 201012:57

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TLDRThe video script explores the concept of finding distinct complex numbers z that satisfy a given determinant equation equaling zero. It begins by defining omega as a complex number derived from cosine and sine functions. The script then meticulously evaluates a 3x3 determinant, simplifies the resulting polynomial equation, and applies Euler's formula to express omega in exponential form. Through a series of algebraic manipulations, it is revealed that z^3 = 0 is the simplified equation, leading to the conclusion that there is only one complex number, z = 0, satisfying the original determinant condition.

Takeaways

🧮 The problem involves finding the number of distinct complex numbers z that satisfy a determinant equation equal to zero, where z is a complex number expressed in terms of cosine and sine functions.
📏 The determinant in question is a 3x3 matrix, and the process involves evaluating this determinant to solve for z.
🔍 The determinant simplification follows a checkerboard pattern, which is a common method for calculating determinants of larger matrices by breaking them down into smaller submatrices.
✖️ The script involves multiplying out terms and simplifying expressions involving powers of z and omega, which is defined as a complex number e^(2πi/3).
🤔 The script uses Euler's formula to express omega in terms of exponentials, which simplifies the calculation of its powers.
📐 The unit circle is used to find the values of cosine and sine for the angle 2π/3, which helps in expressing omega and its powers.
🔢 Powers of omega are calculated by considering the angle's rotation on the unit circle and its corresponding values on the complex plane.
➗ Throughout the process, terms are systematically canceled out due to their values being additive inverses of each other.
🧐 The final simplified form of the determinant is found to be z^3 = 0, which implies that the only solution for z is zero.
🎯 The question is not about the solution for z, but rather the number of distinct complex numbers that satisfy the determinant equation, which, in this case, is just one: z = 0.
📝 The entire process is a detailed walkthrough of evaluating a determinant, factoring in complex numbers, and applying trigonometric and exponential identities.

Q & A

What is the value of omega in the context of the script?
-Omega is defined as the complex number cosine of 2π/3 plus i sine of 2π/3.
What is the determinant equation being solved for in the script?
-The determinant equation being solved for is a 3x3 matrix with the determinant set to equal 0, and the variable z is to be solved or its properties determined.
How does the determinant of the sub-2x2 matrix relate to the overall determinant equation?
-The determinant of the sub-2x2 matrix is multiplied by (z + 1) and combined with other terms to form the overall determinant equation.
What is the significance of the checkerboard pattern when evaluating determinants?
-The checkerboard pattern is used to alternate signs when expanding the determinant, which helps in simplifying the calculation.
How does the script simplify the determinant equation?
-The script simplifies the determinant equation by multiplying out terms, canceling out like terms, and grouping terms by the powers of z.
What is Euler's formula mentioned in the script, and how is it used?
-Euler's formula states that e^(iθ) = cos(θ) + i*sin(θ). It is used to rewrite omega as e^(2πi/3) to simplify the powers of omega in the determinant equation.
What is the unit circle and how does it relate to the powers of omega?
-The unit circle is a circle with a radius of 1, centered at the origin, used to represent complex numbers in the complex plane. It is used to find the values of omega and its powers by identifying their positions on the unit circle corresponding to angles of 2π/3, 4π/3, 2π, etc.
What are the values of omega squared, omega cubed, and omega to the fourth power as per the script?
-Omega squared is -1/2 - sqrt(3)i/2, omega cubed is 1, and omega to the fourth power is -1/2 + sqrt(3)i/2.
How does the script conclude the number of distinct complex numbers z that satisfy the determinant equation?
-The script concludes that z^3 = 0 is the simplified form of the determinant equation, and since the only number whose cube is zero is z = 0, there is only one distinct complex number that satisfies the equation.
What is the final answer to the number of distinct complex numbers z that satisfy the given determinant equation?
-The final answer is that there is only 1 distinct complex number, z = 0, that satisfies the given determinant equation.

Outlines

00:00

🧮 Complex Numbers and Determinants

The paragraph introduces a complex number problem involving omega, defined as cos(2π/3) + i*sin(2π/3). The task is to find the number of distinct complex solutions z for a given 3x3 determinant equation set to zero. The explanation begins by evaluating the determinant, breaking it down into sub-determinants, and applying the properties of determinants to simplify the equation. The process involves multiplying and simplifying terms, considering the checkerboard pattern of signs in determinant expansion, and eventually aiming to solve for z.

05:00

🔍 Simplifying the Determinant Equation

This section continues the exploration of the determinant equation, focusing on simplifying the expression by grouping like terms of z to different powers. The speaker multiplies terms out, simplifies, and cancels out terms where possible. The goal is to express the determinant equation in a simplified form that can be more easily analyzed to find the complex numbers z that satisfy the equation. The process involves careful algebraic manipulation and an understanding of complex number properties.

10:01

📚 Euler's Formula and Powers of Omega

The final paragraph utilizes Euler's formula to express omega in exponential form, which simplifies the process of raising omega to various powers. The speaker evaluates omega, omega squared, omega cubed, and omega to the fourth and sixth powers using the unit circle and trigonometric identities. The simplification reveals that many terms in the determinant equation cancel out, leading to the conclusion that z^3 = 0 is the simplified form of the equation. This implies that the only solution for z is zero, and thus, there is only one distinct complex number that satisfies the original determinant equation.

Mindmap

Keywords

💡Complex Number

A complex number is a number that can be expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit with the property that i^2 = -1. In the context of the video, complex numbers are used to define the variable 'omega' and to explore the solutions to a determinant equation. The script introduces omega as a complex number in the form of cosine plus i times sine, which is a standard way to represent complex numbers using trigonometric functions.

💡Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. In the video, the determinant of a 3x3 matrix is set to zero to find the values of the complex number 'z' that satisfy the equation. The process of evaluating the determinant is central to the script's exploration of the solutions to the given mathematical problem.

💡Submatrix

A submatrix is a smaller matrix derived from a larger matrix by removing some of its rows and/or columns. In the script, the term 'sub-2 by 2 matrix' refers to a smaller 2x2 matrix that is part of the larger 3x3 matrix whose determinant is being calculated. The determinant of the submatrix is used in the process of expanding the determinant of the larger matrix.

💡Euler's Formula

Euler's formula states that e^(iθ) = cos(θ) + i*sin(θ), which is a fundamental equation in complex analysis that establishes a deep relationship between trigonometry and the complex exponential function. In the video, Euler's formula is used to express the complex number 'omega' as e^(2πi/3), simplifying the powers of 'omega' and facilitating the solution to the determinant equation.

💡Unit Circle

The unit circle is a circle with a radius of one, centered at the origin of a coordinate system. It is a fundamental concept in trigonometry and complex number representation. The script refers to the unit circle when determining the values of cosine and sine for specific angles, such as 2π/3, which are used to define the complex number 'omega'.

💡Trigonometric Functions

Trigonometric functions, such as sine and cosine, are mathematical functions that relate the angles of a triangle to the lengths of its sides. In the video, these functions are used to define the complex number 'omega' and to calculate its powers, which are essential for simplifying the determinant equation.

💡Exponential Function

The exponential function with a complex argument, such as e^(iθ), is a way to represent complex numbers in polar form. In the script, the exponential function is used to rewrite 'omega' and its powers, making it easier to handle the complex arithmetic involved in solving the determinant equation.

💡Powers of a Complex Number

Raising a complex number to a power involves multiplying the number by itself that many times. In the video, the powers of 'omega' (omega squared, cubed, etc.) are calculated to simplify the determinant equation. The script shows how these powers can be simplified using Euler's formula and the properties of the unit circle.

💡Simplification

Simplification in mathematics refers to the process of making an expression or equation easier to understand or solve by reducing it to its simplest form. The script demonstrates the simplification of complex expressions involving powers of 'omega' and the determinant equation, which ultimately leads to the conclusion that there is only one solution for 'z'.

💡Distinct Complex Numbers

Distinct complex numbers refer to complex numbers that are different from each other. In the context of the video, the script concludes that there is only one distinct complex number 'z' that satisfies the given determinant equation, which is 'z' equals zero.

Highlights

Let omega be the complex number cosine of 2 pi over 3, plus i sine of 2 pi over 3

Evaluate the 3 by 3 determinant equaling 0 to solve for z

Simplify the determinant using the checkerboard pattern

Express the determinant as a cubic polynomial in z

Simplify the polynomial by grouping like terms

Use Euler's formula to rewrite omega as e^(2 pi i / 3)

Compute powers of omega using Euler's formula and the unit circle

Simplify the polynomial further by substituting the values of omega, omega^2, omega^3, etc.

The simplified polynomial is z^3 = 0

The only solution is z = 0

There is only 1 distinct complex number z satisfying the determinant equation

The innovative method involves evaluating determinants and using complex numbers

The theoretical contribution is solving for z using a cubic polynomial

The practical application is finding the number of solutions to a determinant equation

Euler's formula and the unit circle are key to simplifying the expression

The process demonstrates the power of algebraic manipulation and substitution

The solution involves multiple steps of simplification and substitution

The final answer is obtained by recognizing that z^3 = 0 implies z = 0

The method can be applied to other similar determinant equations

Transcripts

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Takeaways

Q & A

What is the value of omega in the context of the script?

What is the determinant equation being solved for in the script?

How does the determinant of the sub-2x2 matrix relate to the overall determinant equation?

What is the significance of the checkerboard pattern when evaluating determinants?

How does the script simplify the determinant equation?

What is Euler's formula mentioned in the script, and how is it used?

What is the unit circle and how does it relate to the powers of omega?

What are the values of omega squared, omega cubed, and omega to the fourth power as per the script?

How does the script conclude the number of distinct complex numbers z that satisfy the determinant equation?

What is the final answer to the number of distinct complex numbers z that satisfy the given determinant equation?