Multiplying a matrix by a matrix | Matrices | Precalculus | Khan Academy

Khan Academy
6 Aug 201305:29
EducationalLearning
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TLDRThe video script explains the process of matrix multiplication, emphasizing the importance of compatible dimensions for the operation to be valid. It demonstrates the multiplication of a 2x3 matrix (E) by a 3x2 matrix (D), resulting in a 2x2 matrix. The script details each step of the computation, highlighting that the order of multiplication matters and that matrix multiplication is not commutative. The final product is a 2x2 matrix with calculated values, confirming the correctness of the procedure.

Takeaways
  • 📌 Matrix multiplication is a human-defined operation with specific rules.
  • 🔢 The number of columns in the first matrix must equal the number of rows in the second matrix for multiplication to be valid.
  • 🔄 The order of matrix multiplication matters; E × D is not necessarily the same as D × E.
  • 🤔 The product ED results in a matrix with dimensions (rows of E) by (columns of D).
  • 🧠 Understanding the concept of dot product is helpful when multiplying matrices.
  • 📍 Each entry in the resulting matrix is computed by a dot product of a row from the first matrix and a column from the second matrix.
  • 🔢 The top left entry of the product is calculated as (0×3) + (3×3) + (5×4).
  • 🔢 The top right entry is calculated as (0×4) + (3×-2) + (5×-2).
  • 🔢 The bottom left entry is calculated as (5×3) + (5×3) + (2×4).
  • 🔢 The bottom right entry is calculated as (5×4) + (5×-2) + (2×-2).
  • 📋 The final resulting matrix from ED is [29, -16; 38, 6].
Q & A

  • What are the dimensions of matrix E?

    -Matrix E has two rows and three columns, making it a 2x3 matrix.

  • What are the dimensions of matrix D?

    -Matrix D has three rows and two columns, making it a 3x2 matrix.

  • Is the matrix multiplication ED valid? Why or why not?

    -Yes, the matrix multiplication ED is valid because the number of columns in matrix E (3) is equal to the number of rows in matrix D (3), which is a requirement for matrix multiplication.

  • What is the significance of the order of matrices in multiplication?

    -The order of matrices in multiplication matters because matrix multiplication is not commutative. ED is not necessarily equal to DE. The specific order determines the resulting matrix's dimensions and values.

  • What is the resulting matrix of the multiplication ED?

    -The resulting matrix of the multiplication ED is a 2x2 matrix.

  • How is the top left entry of the resulting matrix calculated?

    -The top left entry is calculated by multiplying the first row of matrix E by the first column of matrix D and summing the products: (0 * 3) + (3 * 3) + (5 * 4).

  • What is the calculation for the top right entry of the resulting matrix?

    -The top right entry is calculated by multiplying the first row of matrix E by the second column of matrix D and summing the products: (0 * 4) + (3 * -2) + (5 * -2).

  • How do you compute the bottom left entry of the resulting matrix?

    -The bottom left entry is computed by multiplying the second row of matrix E by the first column of matrix D and summing the products: (5 * 3) + (5 * 3) + (2 * 4).

  • What is the calculation for the bottom right entry of the resulting matrix?

    -The bottom right entry is calculated by multiplying the second row of matrix E by the second column of matrix D and summing the products: (5 * 4) + (5 * -2) + (2 * -2).

  • What is the final resulting matrix after performing the multiplication ED?

    -The final resulting matrix after performing the multiplication ED is [29, -16; 38, 6].

  • Why is it important to check the dimensions before performing matrix multiplication?

    -It is important to check the dimensions before performing matrix multiplication to ensure that the operation is valid. The number of columns in the first matrix must be equal to the number of rows in the second matrix for the multiplication to be defined.

Outlines
00:00
📝 Matrix Multiplication Process

This paragraph explains the process of matrix multiplication, focusing on the validity of the operation and the importance of order. It begins with the identification of two matrices, E and D, and the question of their product, ED. The speaker then checks if the matrices can be multiplied based on the alignment of rows and columns. The explanation emphasizes that the number of columns in the first matrix must equal the number of rows in the second matrix for the operation to be valid. The paragraph also highlights that matrix multiplication is not commutative, meaning E times D is different from D times E. The speaker proceeds to calculate the product, detailing each step of the computation, including the dot product of rows and columns to find the entries of the resulting matrix. The process is shown for a 2x3 matrix multiplied by a 3x2 matrix, resulting in a 2x2 matrix. The computation involves multiplying and adding corresponding elements from rows of the first matrix and columns of the second matrix. The final result is a 2x2 matrix with entries calculated through this method.

05:02
🔢 Result Verification of Matrix Multiplication

In this paragraph, the speaker verifies the result of the matrix multiplication ED discussed in the previous section. The resulting 2x2 matrix is presented with its computed values, which are 29, -16, 38, and 6. The speaker confirms the correctness of the calculated matrix, indicating a successful completion of the matrix multiplication process. This paragraph serves as a conclusion to the matrix multiplication example, providing the final outcome and ensuring the accuracy of the method used.

Mindmap
Keywords
💡Matrices
Matrices are rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns. In the video, matrix E and matrix D are given as examples of matrices, which are the objects of interest for the multiplication operation being discussed. The concept of matrices is fundamental to the video's theme as it explores the process of matrix multiplication, a key operation in linear algebra with applications in various fields such as computer graphics, engineering, and data analysis.
💡Matrix Multiplication
Matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. The video emphasizes that this operation is defined under the condition that the number of columns in the first matrix is equal to the number of rows in the second matrix. It is a crucial concept in the video, as it is the main operation being performed and explained in detail. The order of multiplication matters, as E times D is not necessarily the same as D times E, a property known as non-commutativity.
💡Rows and Columns
Rows and columns are the fundamental structure of a matrix. A matrix is composed of horizontal lines (rows) and vertical lines (columns), where each intersection point is an element of the matrix. In the context of the video, the dimensions of the matrices (rows and columns) determine the validity of the matrix multiplication operation. The video uses the terms 'rows' and 'columns' to describe the structure of matrix E and matrix D, which are essential for understanding the process of matrix multiplication.
💡Dot Product
The dot product is a mathematical operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. In the context of the video, the dot product is used to calculate the individual elements of the resulting matrix after matrix multiplication. The video explains that the dot product is taken between a row of the first matrix and a column of the second matrix to find the corresponding entry in the product matrix.
💡Scratch Pad
A scratch pad, in the context of the video, refers to a workspace where calculations can be performed. It is a place for temporary notes and computations, which is essential when dealing with complex mathematical operations like matrix multiplication. The video mentions getting out a 'little scratch pad' to work out the matrix multiplication, indicating the need for a space to organize and perform the calculations.
💡2 by 2 Matrix
A 2 by 2 matrix is a matrix that has 2 rows and 2 columns, containing a total of 4 elements. In the video, the result of the matrix multiplication ED is a 2 by 2 matrix, which is a key outcome of the operation being explained. The concept is important as it sets the expectation for the dimensions of the output when two matrices are multiplied, and it is directly related to the calculations performed in the video.
💡Valid Operation
A valid operation, in the context of matrix multiplication, refers to an operation that can be performed according to the rules defined for matrices. In the video, the speaker checks whether the multiplication of matrix E by matrix D is a valid operation by ensuring the number of columns in the first matrix matches the number of rows in the second matrix. This concept is crucial as it determines whether the multiplication can proceed and is a fundamental rule in matrix algebra.
💡Order of Multiplication
The order of multiplication refers to the sequence in which matrices are multiplied. In the context of the video, it is emphasized that the order matters when multiplying matrices, meaning that the result of E times D can be different from D times E. This concept is important as it highlights a key property of matrix multiplication that differs from the multiplication of scalar quantities, where the order does not affect the result.
💡Computation
Computation refers to the process of performing mathematical calculations to find a solution or reach a result. In the video, the term is used in the context of working out the product of two matrices, which involves a series of computations to determine each element of the resulting matrix. The video walks through the computations step by step, demonstrating how to perform the matrix multiplication and calculate the final product.
💡Dimensions
Dimensions in the context of matrices refer to the size of the matrix, specifically the number of rows and columns it contains. In the video, the dimensions of the matrices are crucial for determining whether matrix multiplication is possible and for predicting the size of the resulting matrix. The script mentions the dimensions of matrix E and matrix D to explain the process of multiplying them and to calculate the dimensions of the product matrix.
💡Non-Commutativity
Non-commutativity is a property of certain mathematical operations, including matrix multiplication, which means that changing the order of the operands can change the result. In the video, this property is mentioned to highlight that E times D is not necessarily the same as D times E, which is an important concept to understand when working with matrices and their multiplication.
Highlights

Matrix multiplication is a human-defined operation with specific rules.

Matrix multiplication is only valid when the number of columns in the first matrix equals the number of rows in the second matrix.

The order of matrix multiplication matters, unlike with regular numbers.

The resulting matrix from E x D will have the same number of rows as the first matrix and the same number of columns as the second matrix.

Each entry in the resulting matrix is computed by taking the dot product of the corresponding row from the first matrix and the column from the second matrix.

The top left entry of the product matrix is calculated as (0 * 3) + (3 * 3) + (5 * 4).

The top right entry is calculated by using the first row of the first matrix and the second column of the second matrix.

The bottom left entry is found by multiplying the second row of the first matrix with the first column of the second matrix.

The bottom right entry involves the second row of the first matrix and the second column of the second matrix.

The resulting matrix from E x D is a 2 by 2 matrix with entries calculated through a series of multiplications and additions.

The final product matrix is 29, -16, 38, 6.

Matrix multiplication requires verifying the validity of the operation based on the dimensions of the matrices involved.

The process of matrix multiplication involves a series of dot products and is not commutative.

The example provided demonstrates the step-by-step process of multiplying two matrices with specific numerical values.

The transcript provides a detailed explanation of the matrix multiplication process, including the importance of order and the computation of each matrix entry.

Transcripts

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MathematicsMatrix MultiplicationEducationalLinear AlgebraMath TutorialDot ProductMathematics EducationMatrix TheoryOperation OrderDimension Compatibility