Step 1 Of 3Given:${ A=\left(\begin{array}{ll} 2 & 4 \ 1 & 3 \end{array}\right) \quad B=\left(\begin{array}{cc} 1 & 2 \ 3 & -1 \end{array}\right) }$Calculate The Product Of Matrices { A \cdot B $} . . . [ A \cdot B = \square

Introduction

Matrix multiplication is a fundamental concept in linear algebra, and it has numerous applications in various fields, including physics, engineering, computer science, and data analysis. In this article, we will focus on calculating the product of two matrices, A and B, using the given matrices.

Given Matrices

We are given two matrices, A and B, as follows:

${ A=\left(\begin{array}{ll} 2 & 4 \\ 1 & 3 \end{array}\right) \quad B=\left(\begin{array}{cc} 1 & 2 \\ 3 & -1 \end{array}\right) }$

Matrix Multiplication Rules

Before we proceed with the calculation, let's recall the rules of matrix multiplication:

• The number of columns in the first matrix must be equal to the number of rows in the second matrix.
• The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.
• Each element of the resulting matrix is calculated by multiplying the corresponding elements of a row in the first matrix with the corresponding elements of a column in the second matrix and summing the products.

Calculating the Product of Matrices A and B

Now, let's calculate the product of matrices A and B using the given matrices.

To calculate the product of matrices A and B, we need to multiply the corresponding elements of each row in matrix A with the corresponding elements of each column in matrix B and sum the products.

The resulting matrix will have the same number of rows as matrix A and the same number of columns as matrix B.

Let's calculate the elements of the resulting matrix:

• Element (1,1) = (2)(1) + (4)(3) = 2 + 12 = 14
• Element (1,2) = (2)(2) + (4)(-1) = 4 - 4 = 0
• Element (2,1) = (1)(1) + (3)(3) = 1 + 9 = 10
• Element (2,2) = (1)(2) + (3)(-1) = 2 - 3 = -1

Therefore, the product of matrices A and B is:

${ A \cdot B = \left(\begin{array}{cc} 14 & 0 \\ 10 & -1 \end{array}\right) }$

Conclusion

In this article, we have calculated the product of two matrices, A and B, using the given matrices. We have followed the rules of matrix multiplication and calculated the elements of the resulting matrix. The resulting matrix has the same number of rows as matrix A and the same number of columns as matrix B.

Matrix Addition and Subtraction

Matrix addition and subtraction are also fundamental concepts in linear algebra. In matrix addition, we add corresponding elements of two matrices, while in matrix subtraction, we subtract corresponding elements of two matrices.

Matrix Addition Rules

The rules of matrix addition are as follows:

• The number of rows and columns in both matrices must be the same.
• The resulting matrix will have the same number of rows and columns as the original matrices.
• Each element of the resulting matrix is calculated by adding the corresponding elements of the two matrices.

Matrix Subtraction Rules

The rules of matrix subtraction are as follows:

• The number of rows and columns in both matrices must be the same.
• The resulting matrix will have the same number of rows and columns as the original matrices.
• Each element of the resulting matrix is calculated by subtracting the corresponding elements of the two matrices.

Calculating the Sum of Matrices A and B

Let's calculate the sum of matrices A and B:

${ A + B = \left(\begin{array}{ll} 2 & 4 \\ 1 & 3 \end{array}\right) + \left(\begin{array}{cc} 1 & 2 \\ 3 & -1 \end{array}\right) }$

To calculate the sum of matrices A and B, we need to add the corresponding elements of each row in matrix A with the corresponding elements of each column in matrix B.

The resulting matrix will have the same number of rows and columns as the original matrices.

Let's calculate the elements of the resulting matrix:

• Element (1,1) = (2) + (1) = 3
• Element (1,2) = (4) + (2) = 6
• Element (2,1) = (1) + (3) = 4
• Element (2,2) = (3) + (-1) = 2

Therefore, the sum of matrices A and B is:

${ A + B = \left(\begin{array}{cc} 3 & 6 \\ 4 & 2 \end{array}\right) }$

Calculating the Difference of Matrices A and B

Let's calculate the difference of matrices A and B:

${ A - B = \left(\begin{array}{ll} 2 & 4 \\ 1 & 3 \end{array}\right) - \left(\begin{array}{cc} 1 & 2 \\ 3 & -1 \end{array}\right) }$

To calculate the difference of matrices A and B, we need to subtract the corresponding elements of each row in matrix A with the corresponding elements of each column in matrix B.

The resulting matrix will have the same number of rows and columns as the original matrices.

Let's calculate the elements of the resulting matrix:

• Element (1,1) = (2) - (1) = 1
• Element (1,2) = (4) - (2) = 2
• Element (2,1) = (1) - (3) = -2
• Element (2,2) = (3) - (-1) = 4

Therefore, the difference of matrices A and B is:

${ A - B = \left(\begin{array}{cc} 1 & 2 \\ -2 & 4 \end{array}\right) }$

Conclusion

In this article, we have calculated the sum and difference of two matrices, A and B, using the given matrices. We have followed the rules of matrix addition and subtraction and calculated the elements of the resulting matrices. The resulting matrices have the same number of rows and columns as the original matrices.

Matrix Transpose

The transpose of a matrix is an operator which can be thought of as "swapping" the rows and columns for a matrix.

Matrix Transpose Rules

The rules of matrix transpose are as follows:

• The transpose of a matrix is denoted by the symbol T.
• The transpose of a matrix is obtained by interchanging the rows and columns of the original matrix.
• The transpose of a matrix is also a matrix.

Calculating the Transpose of Matrix A

Let's calculate the transpose of matrix A:

${ A = \left(\begin{array}{ll} 2 & 4 \\ 1 & 3 \end{array}\right) }$

To calculate the transpose of matrix A, we need to interchange the rows and columns of the original matrix.

The resulting matrix will have the same number of rows and columns as the original matrix.

Let's calculate the elements of the resulting matrix:

• Element (1,1) = (2)
• Element (1,2) = (4)
• Element (2,1) = (1)
• Element (2,2) = (3)

Therefore, the transpose of matrix A is:

${ A^T = \left(\begin{array}{cc} 2 & 1 \\ 4 & 3 \end{array}\right) }$

Conclusion

In this article, we have calculated the transpose of a matrix, A, using the given matrix. We have followed the rules of matrix transpose and calculated the elements of the resulting matrix. The resulting matrix has the same number of rows and columns as the original matrix.

Conclusion

In this article, we have covered the basics of matrix multiplication, including the rules of matrix multiplication, calculating the product of matrices, and calculating the sum and difference of matrices. We have also covered the concept of matrix transpose and calculated the transpose of a matrix. The resulting matrices have the same number of rows and columns as the original matrices.

References

• "Linear Algebra and Its Applications" by Gilbert Strang
• "Matrix Algebra" by James E. Gentle
• "Introduction to Linear Algebra" by Gilbert Strang

Further Reading

• "Linear Algebra and Its Applications" by Gilbert Strang
• "Matrix Algebra" by James E. Gentle
• "Introduction to Linear Algebra" by Gilbert Strang

Final Answer

The final answer is:

${ A \cdot B = \left(\begin{array}{cc} 14 & 0 \\ 10 & -1 \end{array}\right) }$

Note

Q&A: Frequently Asked Questions

Q: What is matrix multiplication?

A: Matrix multiplication is a mathematical operation that takes two matrices and produces another matrix. It is a fundamental concept in linear algebra and has numerous applications in various fields, including physics, engineering, computer science, and data analysis.

Q: What are the rules of matrix multiplication?

A: The rules of matrix multiplication are as follows:

• The number of columns in the first matrix must be equal to the number of rows in the second matrix.
• The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.
• Each element of the resulting matrix is calculated by multiplying the corresponding elements of a row in the first matrix with the corresponding elements of a column in the second matrix and summing the products.

Q: How do I calculate the product of two matrices?

A: To calculate the product of two matrices, you need to follow these steps:

1. Check if the number of columns in the first matrix is equal to the number of rows in the second matrix. If not, the matrices cannot be multiplied.
2. Create a new matrix with the same number of rows as the first matrix and the same number of columns as the second matrix.
3. Multiply the corresponding elements of each row in the first matrix with the corresponding elements of each column in the second matrix and sum the products.
4. Place the results in the corresponding positions in the new matrix.

Q: What is the difference between matrix multiplication and matrix addition?

A: Matrix multiplication and matrix addition are two different mathematical operations. Matrix multiplication takes two matrices and produces another matrix by multiplying the corresponding elements of each row in the first matrix with the corresponding elements of each column in the second matrix and summing the products. Matrix addition takes two matrices and produces another matrix by adding the corresponding elements of each row in the first matrix with the corresponding elements of each column in the second matrix.

Q: Can I multiply a matrix by a scalar?

A: Yes, you can multiply a matrix by a scalar. To do this, you need to multiply each element of the matrix by the scalar.

Q: What is the transpose of a matrix?

A: The transpose of a matrix is an operator which can be thought of as "swapping" the rows and columns for a matrix.

Q: How do I calculate the transpose of a matrix?

A: To calculate the transpose of a matrix, you need to interchange the rows and columns of the original matrix.

Q: What are the applications of matrix multiplication?

A: Matrix multiplication has numerous applications in various fields, including physics, engineering, computer science, and data analysis. Some of the applications of matrix multiplication include:

• Linear transformations
• Linear equations
• Systems of linear equations
• Matrix inversion
• Determinants

Q: What are the limitations of matrix multiplication?

A: Matrix multiplication has some limitations. Some of the limitations of matrix multiplication include:

• The number of columns in the first matrix must be equal to the number of rows in the second matrix.
• The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.
• Matrix multiplication is not commutative, meaning that the order of the matrices matters.

Conclusion

In this article, we have covered the basics of matrix multiplication, including the rules of matrix multiplication, calculating the product of matrices, and calculating the sum and difference of matrices. We have also covered the concept of matrix transpose and calculated the transpose of a matrix. The resulting matrices have the same number of rows and columns as the original matrices.

References

• "Linear Algebra and Its Applications" by Gilbert Strang
• "Matrix Algebra" by James E. Gentle
• "Introduction to Linear Algebra" by Gilbert Strang

Further Reading

• "Linear Algebra and Its Applications" by Gilbert Strang
• "Matrix Algebra" by James E. Gentle
• "Introduction to Linear Algebra" by Gilbert Strang

Final Answer

The final answer is:

${ A \cdot B = \left(\begin{array}{cc} 14 & 0 \\ 10 & -1 \end{array}\right) }$

Note

The final answer is the product of matrices A and B.