Is matrix multiplication commutative | Matrices | Precalculus | Khan Academy

Khan Academy

11 Apr 201407:32

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TLDRThe video script explores the concept of commutativity in mathematics, specifically comparing scalar multiplication with matrix multiplication. It explains that while scalar multiplication is commutative, matrix multiplication is not necessarily so. The script uses examples of different matrix dimensions and 2x2 matrices to illustrate that the order of multiplication affects the result, demonstrating that matrix multiplication does not adhere to the commutative property.

Takeaways

📈 Scalar multiplication is commutative, meaning the order of the factors does not affect the result.
🤔 The video explores whether the commutative property applies to matrix multiplication.
🧐 Matrix multiplication may not always be commutative, as the order of multiplication can lead to different results.
🔢 The dimensions of matrices play a crucial role in determining if their multiplication is defined.
🌐 A matrix multiplication is defined if the number of columns in the first matrix equals the number of rows in the second matrix.
🔄 Swapping the order of matrix multiplication does not necessarily result in a defined operation if the matrices are not square or do not have compatible dimensions.
📏 The resulting matrix dimensions from AB and BA can be different, which further illustrates that matrix multiplication is not inherently commutative.
🔢 The script provides a specific example with 2x2 matrices to demonstrate that changing the order of multiplication leads to different matrices.
🚫 Even when both products are defined, the resulting matrices from AB and BA can be distinct, reinforcing the non-commutative nature of matrix multiplication.
📝 Understanding the properties of matrix multiplication is essential for working with linear algebra and its applications.
💡 The video encourages viewers to actively engage with the material and consider the implications of non-commutativity in matrix operations.

Q & A

What is the commutative property of multiplication for scalar quantities?
-The commutative property of multiplication for scalar quantities states that changing the order of the numbers being multiplied does not change the product. For example, 5 times 7 is the same as 7 times 5.
Is the commutative property applicable to matrix multiplication?
-The commutative property is generally not applicable to matrix multiplication. The order in which matrices are multiplied can lead to different results or even make the multiplication undefined.
What are the dimensions of the resulting matrix when multiplying a 5x2 matrix by a 2x3 matrix?
-When multiplying a 5x2 matrix by a 2x3 matrix, the resulting matrix, denoted as C, will have dimensions of 5x3 because the number of columns in the first matrix (2) must match the number of rows in the second matrix (2) for the multiplication to be defined.
Why is the multiplication of a 2x3 matrix by a 5x2 matrix not defined?
-The multiplication of a 2x3 matrix by a 5x2 matrix is not defined because the number of columns in the first matrix (3) does not match the number of rows in the second matrix (5). Matrix multiplication requires that the number of columns of the first matrix be equal to the number of rows of the second matrix.
How does the order of multiplication affect the product of two 2x2 matrices?
-The order of multiplication affects the product of two 2x2 matrices because matrix multiplication is not commutative. Multiplying the first matrix by the second will yield a different result than multiplying the second matrix by the first.
What is the product of the matrix [1, 2; -3, -4] multiplied by the matrix [-2, 0; 0, -3]?
-The product of the matrix [1, 2; -3, -4] and the matrix [-2, 0; 0, -3] is the matrix [-2, -4; 6, 12].
What is the product of the matrix [-2, 0; 0, -3] multiplied by the matrix [1, 2; -3, -4]?
-The product of the matrix [-2, 0; 0, -3] and the matrix [1, 2; -3, -4] is the matrix [-2, -4; 6, 12].
How does the size of the matrices affect the definition of their multiplication?
-The size of the matrices affects the definition of their multiplication because for two matrices to be multiplied, the number of columns in the first matrix must be equal to the number of rows in the second matrix. If this condition is not met, the multiplication is not defined.
What is the significance of the non-commutative property of matrix multiplication?
-The non-commutative property of matrix multiplication is significant because it means that the order in which matrices are combined can drastically change the result. This has important implications in fields like linear algebra, systems of equations, and computer graphics, where the order of operations can lead to different outcomes.
How can one determine if the product of two matrices is defined?
-To determine if the product of two matrices is defined, one must check if the number of columns in the first matrix is equal to the number of rows in the second matrix. If this condition is met, the product is defined; otherwise, it is not.
What is the result of the first entry in the product of the two 2x2 matrices mentioned in the script?
-The result of the first entry in the product of the two 2x2 matrices is -2, which is calculated by 1 times -2 plus 2 times 0.
In the context of the script, why does the product of the matrices change when the order of multiplication is reversed?
-The product of the matrices changes when the order of multiplication is reversed because matrix multiplication involves a series of dot products between rows and columns of the matrices. Changing the order alters these dot products, leading to a different result.

Outlines

00:00

📚 Introduction to Scalar Multiplication and the Question of Matrix Commutativity

This paragraph introduces the commutative property of scalar multiplication, using simple examples to illustrate the concept. It then poses the central question of the video: whether matrix multiplication shares this commutative property. The speaker encourages viewers to ponder this question before proceeding. The paragraph also sets the stage for discussing the dimensions of matrices and how they affect the definition of matrix multiplication, highlighting that the number of columns in the first matrix must equal the number of rows in the second matrix for the product to be defined.

05:03

🤔 Exploring the Definition and Dimensions of Matrix Multiplication

In this paragraph, the speaker delves deeper into the specifics of matrix multiplication, using a 5 by 2 matrix and a 2 by 3 matrix as examples. It explains that the resulting matrix C from the product AB will be a 5 by 3 matrix, as the rows of A and the columns of B determine the dimensions of the product. The speaker then contrasts this with the product BA, pointing out that it is not defined due to the mismatch in the number of columns and rows between matrices A and B. This discrepancy初步揭示了矩阵乘法不满足交换律。

🧠 Order Matters: A Concrete Example with 2x2 Matrices

The speaker presents a concrete example using 2x2 matrices to further illustrate that matrix multiplication is not commutative. By multiplying two specific matrices and showing the step-by-step process, the paragraph demonstrates that the order of multiplication affects the outcome. Even though both products are defined, they result in different matrices. This example reinforces the idea that unlike scalar multiplication, the order in which matrices are multiplied does matter, and the resulting matrices are not necessarily the same.

Mindmap

Keywords

💡Commutativity

Commutativity is a property of mathematical operations, such as multiplication, where the order of the operands does not affect the result. In the context of this video, the concept is explored to determine whether it applies to matrix multiplication. The video demonstrates that while scalar multiplication is commutative, matrix multiplication is not, as changing the order of the matrices can lead to different results or even make the operation undefined.

💡Scalar Quantities

Scalar quantities are single numerical values that can be multiplied together in a commutative manner. In the video, scalars are used to introduce the concept of commutativity, highlighting that the order in which you multiply two numbers does not change the outcome. This serves as a basis for exploring whether a similar property exists for matrix multiplication.

💡Matrix Multiplication

Matrix multiplication is a mathematical operation that involves taking the elements of two matrices and using a specific procedure to calculate a third matrix. Unlike scalar multiplication, matrix multiplication is not commutative, meaning the order of the matrices being multiplied matters and can result in different matrices or even render the operation undefined.

💡Dimensions

In the context of matrices, dimensions refer to the number of rows and columns a matrix has. The dimensions of the resulting matrix from a multiplication operation are determined by the dimensions of the matrices being multiplied. The video emphasizes that for matrix multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix.

💡Defined

In mathematics, an operation is said to be 'defined' if it can be carried out according to the rules of the operation. In the context of the video, it refers to whether the multiplication of two matrices is possible. The video explains that matrix multiplication is defined only when the number of columns in the first matrix matches the number of rows in the second matrix.

💡2 by 2 Matrices

A 2 by 2 matrix is a matrix that has two rows and two columns. It is a specific type of square matrix, and its elements can be used in the context of the video to demonstrate the process and properties of matrix multiplication, including the non-commutative nature of the operation.

💡Row and Column

In a matrix, a 'row' is a horizontal sequence of elements, and a 'column' is a vertical sequence of elements. The position of an element in a matrix is typically referred to by its row number and column number. These concepts are fundamental to understanding how matrix multiplication is performed, as each element in the resulting matrix is computed by multiplying and adding products of the corresponding elements from the rows of the first matrix and the columns of the second matrix.

💡Resulting Matrix

The 'resulting matrix' or 'product matrix' is the matrix that is produced when two matrices are multiplied together. The dimensions and elements of the resulting matrix depend on the dimensions and elements of the original matrices. The video emphasizes that the resulting matrix from AB and BA can be different or even undefined, which is a key point in demonstrating that matrix multiplication is not commutative.

💡Non-Commutative

A non-commutative operation is one where the order of the operands affects the result. In the context of this video, it refers to the property of matrix multiplication, where swapping the order of the matrices being multiplied can lead to different outcomes or make the operation undefined, unlike scalar multiplication which is commutative.

💡Pause the Video

Throughout the video script, the viewer is encouraged to 'pause the video' to think about the concepts being discussed. This is a pedagogical technique used to promote active engagement with the material, allowing the viewer to reflect and potentially arrive at conclusions on their own before proceeding with the explanation.

💡Scalar Multiplication

Scalar multiplication is the process of multiplying a scalar (a single number) by a matrix, which results in a new matrix. Every element of the original matrix is multiplied by the scalar. This operation is commutative, meaning the scalar can be multiplied by the matrix in any order. The video contrasts this with matrix multiplication, which is not commutative.

Highlights

Multiplication of scalar quantities is commutative, meaning the order of multiplication does not affect the result.

The video discusses whether the commutative property applies to matrix multiplication.

Matrix A is a 5 by 2 matrix and matrix B is a 2 by 3 matrix, resulting in a 5 by 3 matrix when multiplied (AB).

The product BA is not defined due to mismatch in the number of columns and rows, indicating that matrix multiplication is not necessarily commutative.

The dimensions of matrices play a crucial role in defining the product of matrix multiplication.

A concrete example is given using 2 by 2 matrices to illustrate the concept.

The product of the two 2 by 2 matrices is calculated, showing that the order of multiplication affects the outcome.

The multiplication of matrices is shown to be non-commutative through the example provided, as the products AB and BA are not the same.

The importance of the number of columns in the first matrix matching the number of rows in the second matrix for matrix multiplication is emphasized.

The video encourages viewers to pause and think about the concepts and examples presented, promoting active engagement with the material.

The process of matrix multiplication is detailed, explaining how each entry in the resulting matrix is calculated.

The video highlights the difference in products when the order of multiplication is changed, even when both products are defined.

The conclusion drawn from the examples is that matrix multiplication is not commutative, reinforcing the main question of the video.

The transcript serves as an educational resource for understanding the properties and intricacies of matrix multiplication.

The video provides a clear and methodical explanation of why matrix multiplication does not always preserve the commutative property.

The transcript showcases the application of theoretical knowledge in the context of matrix dimensions and multiplication, offering practical insights into the subject.

Transcripts

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Takeaways

Q & A

What is the commutative property of multiplication for scalar quantities?

Is the commutative property applicable to matrix multiplication?

What are the dimensions of the resulting matrix when multiplying a 5x2 matrix by a 2x3 matrix?

Why is the multiplication of a 2x3 matrix by a 5x2 matrix not defined?

How does the order of multiplication affect the product of two 2x2 matrices?

What is the product of the matrix [1, 2; -3, -4] multiplied by the matrix [-2, 0; 0, -3]?

What is the product of the matrix [-2, 0; 0, -3] multiplied by the matrix [1, 2; -3, -4]?

How does the size of the matrices affect the definition of their multiplication?

What is the significance of the non-commutative property of matrix multiplication?

How can one determine if the product of two matrices is defined?

What is the result of the first entry in the product of the two 2x2 matrices mentioned in the script?

In the context of the script, why does the product of the matrices change when the order of multiplication is reversed?