How To Find The Determinant of a 3x3 Matrix

The Organic Chemistry Tutor
9 Nov 201811:30
EducationalLearning
32 Likes 10 Comments

TLDRThis video tutorial provides a step-by-step guide on calculating the determinant of a 3x3 matrix. It begins by explaining the basic formula, where the determinant is computed through a series of 2x2 matrix determinants derived from the original 3x3 matrix. The process involves alternating signs and eliminating corresponding row and column elements. The video demonstrates the method with two examples, clearly showing how to compute the determinant for each 2x2 matrix and then combine the results to find the final determinant of the 3x3 matrix. It emphasizes the importance of careful calculation to avoid errors and encourages viewers to practice the method.

Takeaways
  • 📝 The video explains the process of finding the determinant of a 3x3 matrix using a step-by-step method.
  • 🔢 Start by identifying the elements in the first column (a1, a2, a3), second column (b1, b2, b3), and third column (c1, c2, c3) of the matrix.
  • 🌟 Calculate the determinant by eliminating the row and column of each element in the first row and forming two 2x2 matrices.
  • 🧮 Use the rule of alternating signs for the elements of the 2x2 matrices: positive for the first element, negative for the second, and so on.
  • 🔍 Evaluate the determinant of each 2x2 matrix by multiplying diagonals and subtracting the product of the second diagonal from the first.
  • 📌 Provide an example matrix with elements (5, -5, 4; 8, 3, -6; 9, -5, -2) and demonstrate the calculation of its determinant.
  • 🔥 Calculate the determinant by multiplying and combining the results of the 2x2 matrices: (-5 * 9 - 8 * 4) - (3 * 9 - 2 * 8) + (7 * 8 - 2 * 6).
  • 📊 The result of the first example is -737, achieved by carefully following the steps and ensuring all operations are correct.
  • 🔎 Offer a second example with the matrix (8, 4, 3; -5, 6, -2; 7, 9, -8) and guide through the process of elimination and calculation.
  • 📝 The determinant of the second matrix is -717, showing the method's consistency and reliability.
  • 🎓 Emphasize the importance of double-checking work to avoid mistakes that could lead to incorrect results.
  • 🚀 Encourage viewers to practice the method on their own and offer support through additional resources like playlists on pre-calculus, chemistry, physics, and calculus.
Q & A

  • What is the topic of the video?

    -The topic of the video is about finding the determinant of a 3x3 matrix.

  • What is the first step in calculating the determinant of a 3x3 matrix?

    -The first step is to identify the elements in the first column, which are a1, a2, a3, and then to form a 2x2 matrix excluding the row and column of the first element.

  • How many 2x2 matrices are formed when calculating the determinant of a 3x3 matrix?

    -Three 2x2 matrices are formed when calculating the determinant, corresponding to the elements in the first row and the elements excluded from the row and column of each element in the first row.

  • What is the formula for the determinant of a 2x2 matrix?

    -The determinant of a 2x2 matrix is found by multiplying the elements of the first diagonal (top-left to bottom-right) and subtracting the product of the elements of the second diagonal (top-right to bottom-left).

  • What is an example 3x3 matrix provided in the video?

    -An example 3x3 matrix provided is [[5, -5, 4], [3, 2, -6], [7, 8, -5]].

  • What is the final determinant calculated for the example 3x3 matrix in the video?

    -The final determinant calculated for the example matrix is -737.

  • How does the process of calculating the determinant ensure accuracy?

    -The process requires double-checking the work to ensure that all elements are correctly identified and excluded from the appropriate rows and columns, as one mistake can lead to an incorrect final answer.

  • What is the second example 3x3 matrix provided in the video?

    -The second example 3x3 matrix provided is [[8, 4, 3], [-5, 6, -2], [7, 9, -8]].

  • What is the final determinant calculated for the second example 3x3 matrix in the video?

    -The final determinant calculated for the second example matrix is -717.

  • What additional resources are available for learning pre-calculus, chemistry, and physics?

    -The video creator has playlists on pre-calculus, chemistry, physics, and calculus, which can be found on their channel for further study and help.

  • How can one support the video creator and access additional educational content?

    -One can support the video creator by subscribing to their channel and accessing the educational playlists on pre-calculus, chemistry, physics, and calculus.

Outlines
00:00
📚 Introduction to Finding the Determinant of a 3x3 Matrix

This paragraph introduces the concept of finding the determinant of a 3x3 matrix. It explains the initial steps, which involve identifying the elements of the matrix (a1, a2, a3 in the first column, b1, b2, b3 in the second column, and c1, c2, c3 in the third column). The process starts by eliminating the first row and first column to form a 2x2 matrix (b2, b3; c2, c3). It then describes how to alternate signs and form two more 2x2 matrices by eliminating the first and second rows/columns (a2, a3; c2, c3 and a2, a3; b2, b3 respectively). The paragraph emphasizes the importance of accurately following the procedure to avoid mistakes that could lead to incorrect results.

05:02
🔢 Calculating the Determinant of a 2x2 Matrix

This paragraph focuses on the method of calculating the determinant of a 2x2 matrix, which is a crucial step in finding the determinant of a 3x3 matrix. It explains the process of multiplying the diagonal elements (b2*c3 and c2*b3) and alternating the signs to get the determinant. The paragraph provides a clear example by calculating the determinant of the 2x2 matrix (-5, 4; 8, 9) and emphasizes the need to double-check the work to ensure accuracy. It also encourages the viewer to practice by finding the determinant of another 3x3 matrix with given elements (5, -5, 4; 8, 6, -2; 7, 9, -8).

10:03
📝 Final Steps and Additional Example

The final paragraph concludes the process of finding the determinant of a 3x3 matrix by providing the remaining calculations for the given example. It details the steps of evaluating the determinant of the second 2x2 matrix (6, 9; -2, -8) and combining the results from the previous steps. The paragraph then presents an additional example with a new 3x3 matrix (8, 4, 3; -5, 6, -2; 7, 9, -8) and outlines the process of finding its determinant. The summary of calculations is provided, leading to the final answer of -717. The paragraph wraps up by encouraging viewers to subscribe to the channel for more content on related subjects.

Mindmap
Keywords
💡determinant
The determinant is a scalar value that can be computed from the elements of a square matrix and is used to find the invertibility of the matrix. In the context of the video, the determinant is the main focus, with the process of calculating it for a 3x3 matrix explained in detail. For instance, the video walks through the steps of eliminating elements in specific rows and columns to create smaller 2x2 matrices, which are then used to compute the determinant of the original 3x3 matrix.
💡three by three matrix
A three by three matrix is a square matrix that contains nine elements arranged in three rows and three columns. In the video, the process of finding the determinant of such a matrix is thoroughly explained. This type of matrix is important because it allows for the application of various mathematical operations and properties, including the calculation of its determinant.
💡two by two matrix
A two by two matrix is a square matrix consisting of two rows and two columns, and it contains four elements. In the video, the process of calculating the determinant of a 3x3 matrix involves creating and evaluating 2x2 matrices as intermediate steps. These smaller matrices are derived by eliminating certain rows and columns from the original 3x3 matrix, and their determinants contribute to the final result.
💡row and column elimination
Row and column elimination is a method used in the process of calculating the determinant of a 3x3 matrix. It involves removing the elements in a specified row and column to create smaller 2x2 matrices. This technique is crucial for simplifying the original matrix and making the determinant calculation manageable. The video script illustrates this by showing how to eliminate the first row and column to form new matrices for determinant calculation.
💡diagonal multiplication
Diagonal multiplication is a technique used in the calculation of the determinant of a 2x2 matrix, where the elements of the diagonal are multiplied with each other. The video explains that the determinant of a 2x2 matrix is found by multiplying the elements that form the two diagonals and then subtracting the product of the second diagonal from the product of the first diagonal.
💡sign alternation
Sign alternation is a pattern used in the calculation of the determinant of a 3x3 matrix, where the sign of the product of elements alternates between positive and negative. This pattern helps in keeping track of the signs as you calculate the determinant by following the rule of a positive sign for the first element in the first row and then alternating signs for the subsequent elements.
💡matrix elements
Matrix elements refer to the individual numbers or values that make up the entries of a matrix. In the context of the video, understanding and manipulating matrix elements is essential for calculating the determinant. The process involves selecting specific elements based on their position in the matrix and using them to form smaller matrices or to contribute directly to the determinant calculation.
💡invertibility
Invertibility in the context of matrices refers to the property of a matrix that allows it to have an inverse. A matrix is invertible if and only if its determinant is non-zero. The determinant is thus a key factor in determining whether a matrix can be reversed, which is important in various mathematical and computational applications.
💡pre-calculus
Pre-calculus is a branch of mathematics that covers topics which are typically studied before calculus. It includes the study of functions, trigonometry, and matrices, among other topics. In the video, the concept of finding the determinant of a matrix is a pre-calculus topic, as it is a foundational skill necessary for understanding more advanced mathematical concepts like linear transformations and eigenvalues, which are covered in pre-calculus courses.
💡mathematical operations
Mathematical operations refer to the various processes or techniques used to perform calculations and manipulate numbers, symbols, or expressions. In the video, several mathematical operations are used in the process of calculating the determinant, such as multiplication, subtraction, and the elimination of matrix elements.
💡video demonstration
A video demonstration is a visual and auditory presentation that walks viewers through a process or procedure. In the context of the video script, the demonstration refers to the step-by-step explanation and illustration of how to find the determinant of a 3x3 matrix. The video format helps viewers understand the process by showing the calculations and providing examples.
Highlights

Introduction to finding the determinant of a 3x3 matrix

Formula for a 3x3 matrix with elements a1, a2, a3, b1, b2, b3, c1, c2, c3

Step-by-step explanation of how to calculate the determinant

Use of a 2x2 matrix within the 3x3 determinant calculation

Example calculation with the matrix elements 5, -5, 4, 8, 9, 2, -6, 8, -5, 4

Emphasis on checking work for accuracy in determinant calculations

Detailed calculation of the determinant for the given example

Final answer of -737 for the first example matrix

Introduction to a second example with the matrix elements 8, 4, 3, -5, 6, -2, 7, 9, -8

Calculation of the determinant for the second example using the same method

Final answer of -717 for the second example matrix

Summary of the process for finding the determinant of a 3x3 matrix

Invitation to subscribe to the channel for more pre-calculus, chemistry, physics, and calculus content

Transcripts

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