Select The Correct Answer.For Which Pair Of Matrices Is $A B \neq B A$?A. $A=\left[\begin{array}{cc}1 & 0 \\ 3 & -2\end{array}\right] \quad B=\left[\begin{array}{cc}7 & 0 \\ 3 & 4\end{array}\right\]B. $A=\left[\begin{array}{cc}1 &

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Introduction

Matrix multiplication is a fundamental operation in linear algebra, used to combine two matrices to form a new matrix. However, unlike scalar multiplication, matrix multiplication is not commutative, meaning that the order of the matrices matters. In this article, we will explore the conditions under which matrix multiplication is non-commutative, and provide examples to illustrate this concept.

What is Matrix Multiplication?

Matrix multiplication is a binary operation that takes two matrices, A and B, and produces a new matrix, C, such that the elements of C are calculated as the dot product of rows of A with columns of B. The number of columns in A must be equal to the number of rows in B, and the resulting matrix C will have the same number of rows as A and the same number of columns as B.

Matrix Multiplication is Not Commutative

The commutative property of matrix multiplication states that the order of the matrices does not matter, i.e., AB = BA. However, this is not always true. In fact, there are many cases where AB ≠ BA. To determine whether matrix multiplication is commutative, we need to examine the properties of the matrices involved.

Properties of Matrices

To determine whether matrix multiplication is commutative, we need to examine the properties of the matrices involved. Specifically, we need to look at the following properties:

  • Invertibility: If a matrix is invertible, it has an inverse that can be used to "undo" the matrix multiplication.
  • Rank: The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix.
  • Determinant: The determinant of a matrix is a scalar value that can be used to determine whether the matrix is invertible.

Example 1: Non-Commutative Matrix Multiplication

Let's consider the following pair of matrices:

A = \left[\begin{array}{cc}1 & 0 \ 3 & -2\end{array}\right] B = \left[\begin{array}{cc}7 & 0 \ 3 & 4\end{array}\right]

To determine whether matrix multiplication is commutative, we need to calculate AB and BA.

AB = \left[\begin{array}{cc}1 & 0 \ 3 & -2\end{array}\right] \left[\begin{array}{cc}7 & 0 \ 3 & 4\end{array}\right] = \left[\begin{array}{cc}7 & 0 \ -9 & -8\end{array}\right]

BA = \left[\begin{array}{cc}7 & 0 \ 3 & 4\end{array}\right] \left[\begin{array}{cc}1 & 0 \ 3 & -2\end{array}\right] = \left[\begin{array}{cc}7 & 0 \ 15 & -8\end{array}\right]

As we can see, AB ≠ BA, which means that matrix multiplication is not commutative for these two matrices.

Example 2: Commutative Matrix Multiplication

Let's consider the following pair of matrices:

A = \left[\begin{array}{cc}1 & 0 \ 0 & 1\end{array}\right] B = \left[\begin{array}{cc}1 & 0 \ 0 & 1\end{array}\right]

To determine whether matrix multiplication is commutative, we need to calculate AB and BA.

AB = \left[\begin{array}{cc}1 & 0 \ 0 & 1\end{array}\right] \left[\begin{array}{cc}1 & 0 \ 0 & 1\end{array}\right] = \left[\begin{array}{cc}1 & 0 \ 0 & 1\end{array}\right]

BA = \left[\begin{array}{cc}1 & 0 \ 0 & 1\end{array}\right] \left[\begin{array}{cc}1 & 0 \ 0 & 1\end{array}\right] = \left[\begin{array}{cc}1 & 0 \ 0 & 1\end{array}\right]

As we can see, AB = BA, which means that matrix multiplication is commutative for these two matrices.

Conclusion

In conclusion, matrix multiplication is not always commutative. The order of the matrices matters, and there are many cases where AB ≠ BA. To determine whether matrix multiplication is commutative, we need to examine the properties of the matrices involved, such as invertibility, rank, and determinant. By understanding the conditions under which matrix multiplication is non-commutative, we can better appreciate the complexities of linear algebra and matrix theory.

References

  • Linear Algebra and Its Applications by Gilbert Strang
  • Matrix Theory by Richard Bellman
  • Introduction to Linear Algebra by Jim Hefferon

Further Reading

  • Matrix Multiplication and Non-Commutativity by Wikipedia
  • Matrix Theory by MathWorld
  • Linear Algebra by Khan Academy
    Matrix Multiplication and Non-Commutativity: Q&A =====================================================

Introduction

Matrix multiplication is a fundamental operation in linear algebra, used to combine two matrices to form a new matrix. However, unlike scalar multiplication, matrix multiplication is not commutative, meaning that the order of the matrices matters. In this article, we will answer some frequently asked questions about matrix multiplication and non-commutativity.

Q: What is matrix multiplication?

A: Matrix multiplication is a binary operation that takes two matrices, A and B, and produces a new matrix, C, such that the elements of C are calculated as the dot product of rows of A with columns of B.

Q: Why is matrix multiplication not commutative?

A: Matrix multiplication is not commutative because the order of the matrices matters. The resulting matrix C will have the same number of rows as A and the same number of columns as B, but the elements of C will be different depending on the order of the matrices.

Q: What are some examples of non-commutative matrix multiplication?

A: Here are a few examples of non-commutative matrix multiplication:

  • A = \left[\begin{array}{cc}1 & 0 \ 3 & -2\end{array}\right] B = \left[\begin{array}{cc}7 & 0 \ 3 & 4\end{array}\right] AB = \left[\begin{array}{cc}7 & 0 \ -9 & -8\end{array}\right] BA = \left[\begin{array}{cc}7 & 0 \ 15 & -8\end{array}\right]
  • A = \left[\begin{array}{cc}1 & 0 \ 0 & 1\end{array}\right] B = \left[\begin{array}{cc}0 & 1 \ 1 & 0\end{array}\right] AB = \left[\begin{array}{cc}0 & 1 \ 1 & 0\end{array}\right] BA = \left[\begin{array}{cc}1 & 0 \ 0 & 1\end{array}\right]

Q: What are some examples of commutative matrix multiplication?

A: Here are a few examples of commutative matrix multiplication:

  • A = \left[\begin{array}{cc}1 & 0 \ 0 & 1\end{array}\right] B = \left[\begin{array}{cc}1 & 0 \ 0 & 1\end{array}\right] AB = \left[\begin{array}{cc}1 & 0 \ 0 & 1\end{array}\right] BA = \left[\begin{array}{cc}1 & 0 \ 0 & 1\end{array}\right]
  • A = \left[\begin{array}{cc}2 & 0 \ 0 & 2\end{array}\right] B = \left[\begin{array}{cc}2 & 0 \ 0 & 2\end{array}\right] AB = \left[\begin{array}{cc}4 & 0 \ 0 & 4\end{array}\right] BA = \left[\begin{array}{cc}4 & 0 \ 0 & 4\end{array}\right]

Q: How can I determine whether matrix multiplication is commutative?

A: To determine whether matrix multiplication is commutative, you can calculate AB and BA and compare the results. If AB = BA, then matrix multiplication is commutative. If AB ≠ BA, then matrix multiplication is not commutative.

Q: What are some real-world applications of matrix multiplication?

A: Matrix multiplication has many real-world applications, including:

  • Computer Graphics: Matrix multiplication is used to perform transformations on 2D and 3D objects, such as rotations, translations, and scaling.
  • Machine Learning: Matrix multiplication is used in machine learning algorithms, such as neural networks and support vector machines.
  • Data Analysis: Matrix multiplication is used in data analysis to perform operations such as matrix multiplication, matrix addition, and matrix subtraction.

Conclusion

In conclusion, matrix multiplication is a fundamental operation in linear algebra that is used to combine two matrices to form a new matrix. However, unlike scalar multiplication, matrix multiplication is not commutative, meaning that the order of the matrices matters. By understanding the conditions under which matrix multiplication is non-commutative, we can better appreciate the complexities of linear algebra and matrix theory.