If $a$ And $b$ Are Nonzero Real Numbers, $A=\begin{bmatrix} A & A \ B & B \end{bmatrix}$, And $ B = [ A A − A − A ] B=\begin{bmatrix} A & A \\ -a & -a \end{bmatrix} B = [ A − A ​ A − A ​ ] [/tex], Find $AB$ And $BA$.Select

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Introduction

In linear algebra, matrix multiplication is a fundamental operation used to combine two matrices to form a new matrix. Given two matrices A and B, the product AB is a matrix whose elements are defined by the dot product of the rows of A with the columns of B. In this article, we will explore the matrix multiplication of two specific matrices A and B, and find their products AB and BA.

Matrix A and Matrix B

Let's consider two matrices A and B, defined as follows:

A=[aabb]A=\begin{bmatrix} a & a \\ b & b \end{bmatrix}

B=[aa−a−a]B=\begin{bmatrix} a & a \\ -a & -a \end{bmatrix}

where a and b are nonzero real numbers.

Finding AB

To find the product AB, we need to multiply the rows of A with the columns of B. The resulting matrix will have the same number of rows as A and the same number of columns as B.

AB=[aabb][aa−a−a]AB=\begin{bmatrix} a & a \\ b & b \end{bmatrix}\begin{bmatrix} a & a \\ -a & -a \end{bmatrix}

Using the rules of matrix multiplication, we can calculate the elements of the resulting matrix:

AB=[a⋅a+a⋅(−a)a⋅a+a⋅(−a)b⋅a+b⋅(−a)b⋅a+b⋅(−a)]AB=\begin{bmatrix} a \cdot a + a \cdot (-a) & a \cdot a + a \cdot (-a) \\ b \cdot a + b \cdot (-a) & b \cdot a + b \cdot (-a) \end{bmatrix}

Simplifying the expressions, we get:

AB=[0000]AB=\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}

Finding BA

To find the product BA, we need to multiply the rows of B with the columns of A. The resulting matrix will have the same number of rows as B and the same number of columns as A.

BA=[aa−a−a][aabb]BA=\begin{bmatrix} a & a \\ -a & -a \end{bmatrix}\begin{bmatrix} a & a \\ b & b \end{bmatrix}

Using the rules of matrix multiplication, we can calculate the elements of the resulting matrix:

BA=[a⋅a+a⋅ba⋅a+a⋅b−a⋅a+(−a)⋅b−a⋅a+(−a)⋅b]BA=\begin{bmatrix} a \cdot a + a \cdot b & a \cdot a + a \cdot b \\ -a \cdot a + (-a) \cdot b & -a \cdot a + (-a) \cdot b \end{bmatrix}

Simplifying the expressions, we get:

BA=[a2+aba2+ab−a2−ab−a2−ab]BA=\begin{bmatrix} a^2 + ab & a^2 + ab \\ -a^2 - ab & -a^2 - ab \end{bmatrix}

Conclusion

In this article, we found the products AB and BA of two specific matrices A and B. We used the rules of matrix multiplication to calculate the elements of the resulting matrices. The product AB was found to be a zero matrix, while the product BA was found to be a matrix with elements involving the variables a and b.

Matrix Multiplication Rules

Matrix multiplication is a fundamental operation in linear algebra, and it has several rules that must be followed. Here are some of the key rules:

  • Rule 1: The number of columns in the first matrix must be equal to the number of rows in the second matrix.
  • Rule 2: The elements of the resulting matrix are calculated by multiplying the rows of the first matrix with the columns of the second matrix.
  • Rule 3: The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix.

Applications of Matrix Multiplication

Matrix multiplication has several applications in various fields, including:

  • Linear Algebra: Matrix multiplication is used to solve systems of linear equations and to find the inverse of a matrix.
  • Computer Graphics: Matrix multiplication is used to perform transformations on 2D and 3D objects.
  • Machine Learning: Matrix multiplication is used in machine learning algorithms, such as neural networks and support vector machines.

Conclusion

In conclusion, matrix multiplication is a fundamental operation in linear algebra, and it has several rules that must be followed. The product AB of two specific matrices A and B was found to be a zero matrix, while the product BA was found to be a matrix with elements involving the variables a and b. Matrix multiplication has several applications in various fields, including linear algebra, computer graphics, and machine learning.

Introduction

In our previous article, we explored the matrix multiplication of two specific matrices A and B, and found their products AB and BA. In this article, we will answer some frequently asked questions about matrix multiplication.

Q: What is matrix multiplication?

A: Matrix multiplication is a fundamental operation in linear algebra that combines two matrices to form a new matrix. The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix.

Q: What are the rules of matrix multiplication?

A: There are three rules of matrix multiplication:

  • Rule 1: The number of columns in the first matrix must be equal to the number of rows in the second matrix.
  • Rule 2: The elements of the resulting matrix are calculated by multiplying the rows of the first matrix with the columns of the second matrix.
  • Rule 3: The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix.

Q: What is the difference between AB and BA?

A: The products AB and BA are not necessarily equal. In our previous article, we found that AB was a zero matrix, while BA was a matrix with elements involving the variables a and b.

Q: When is AB equal to BA?

A: AB is equal to BA when the matrices A and B are square matrices (i.e., they have the same number of rows and columns) and when the elements of the matrices are commutative (i.e., the order of the elements does not matter).

Q: What are the applications of matrix multiplication?

A: Matrix multiplication has several applications in various fields, including:

  • Linear Algebra: Matrix multiplication is used to solve systems of linear equations and to find the inverse of a matrix.
  • Computer Graphics: Matrix multiplication is used to perform transformations on 2D and 3D objects.
  • Machine Learning: Matrix multiplication is used in machine learning algorithms, such as neural networks and support vector machines.

Q: How do I perform matrix multiplication?

A: To perform matrix multiplication, you need to follow the rules of matrix multiplication. Here are the steps:

  1. Check if the matrices are compatible: Make sure that the number of columns in the first matrix is equal to the number of rows in the second matrix.
  2. Calculate the elements of the resulting matrix: Multiply the rows of the first matrix with the columns of the second matrix.
  3. Write the resulting matrix: The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix.

Q: What are some common mistakes to avoid when performing matrix multiplication?

A: Here are some common mistakes to avoid when performing matrix multiplication:

  • Not checking if the matrices are compatible: Make sure that the number of columns in the first matrix is equal to the number of rows in the second matrix.
  • Not following the rules of matrix multiplication: Make sure to multiply the rows of the first matrix with the columns of the second matrix.
  • Not writing the resulting matrix correctly: Make sure to write the resulting matrix with the correct number of rows and columns.

Conclusion

In this article, we answered some frequently asked questions about matrix multiplication. We covered the rules of matrix multiplication, the difference between AB and BA, and the applications of matrix multiplication. We also provided some tips on how to perform matrix multiplication and some common mistakes to avoid.