Multiply The Matrices:${ \left[\begin{array}{cc} 4 & -3 \ 1 & 5 \end{array}\right] \cdot \left[\begin{array}{cc} 3 & 8 \ 1 & 2 \end{array}\right] }$
Introduction
Matrix multiplication is a fundamental concept in linear algebra, and it plays a crucial role in various fields such as physics, engineering, and computer science. In this article, we will focus on multiplying two matrices, and we will provide a step-by-step guide on how to perform this operation.
What is Matrix Multiplication?
Matrix multiplication is a mathematical operation that takes two matrices as input and produces another matrix as output. The resulting matrix is obtained by multiplying the elements of the rows of the first matrix with the elements of the columns of the second matrix.
The Rules of Matrix Multiplication
Before we dive into the example, let's review 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 obtained by multiplying the elements of the corresponding row of the first matrix with the elements of the corresponding column of the second matrix.
Example: Multiplying Two Matrices
Let's consider the following two matrices:
{ \left[\begin{array}{cc} 4 & -3 \\ 1 & 5 \end{array}\right] \cdot \left[\begin{array}{cc} 3 & 8 \\ 1 & 2 \end{array}\right] \}
To multiply these two matrices, we need to follow the rules of matrix multiplication. The resulting matrix will have the same number of rows as the first matrix (2) and the same number of columns as the second matrix (2).
Step 1: Multiply the Elements of the First Row of the First Matrix with the Elements of the First Column of the Second Matrix
The first element of the resulting matrix is obtained by multiplying the elements of the first row of the first matrix with the elements of the first column of the second matrix:
{ (4)(3) + (-3)(1) = 12 - 3 = 9 \}
Step 2: Multiply the Elements of the First Row of the First Matrix with the Elements of the Second Column of the Second Matrix
The second element of the resulting matrix is obtained by multiplying the elements of the first row of the first matrix with the elements of the second column of the second matrix:
{ (4)(8) + (-3)(2) = 32 - 6 = 26 \}
Step 3: Multiply the Elements of the Second Row of the First Matrix with the Elements of the First Column of the Second Matrix
The third element of the resulting matrix is obtained by multiplying the elements of the second row of the first matrix with the elements of the first column of the second matrix:
{ (1)(3) + (5)(1) = 3 + 5 = 8 \}
Step 4: Multiply the Elements of the Second Row of the First Matrix with the Elements of the Second Column of the Second Matrix
The fourth element of the resulting matrix is obtained by multiplying the elements of the second row of the first matrix with the elements of the second column of the second matrix:
{ (1)(8) + (5)(2) = 8 + 10 = 18 \}
The Resulting Matrix
The resulting matrix is:
{ \left[\begin{array}{cc} 9 & 26 \\ 8 & 18 \end{array}\right] \}
Conclusion
In this article, we have provided a step-by-step guide on how to multiply two matrices. We have reviewed the rules of matrix multiplication and have applied them to a specific example. The resulting matrix is a 2x2 matrix, and it is obtained by multiplying the elements of the rows of the first matrix with the elements of the columns of the second matrix.
Applications of Matrix Multiplication
Matrix multiplication has numerous applications in various fields such as physics, engineering, and computer science. Some of the applications of matrix multiplication include:
- Linear Transformations: Matrix multiplication is used to represent linear transformations in mathematics and physics.
- Computer Graphics: Matrix multiplication is used to perform transformations on 2D and 3D objects in computer graphics.
- 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 and matrix inversion.
Final Thoughts
Frequently Asked Questions
In this article, we will answer some of the most frequently asked questions about matrix multiplication.
Q: What is the order of matrix multiplication?
A: The order of matrix multiplication is important. 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 matrices cannot be multiplied.
Q: How do I know if two matrices can be multiplied?
A: To determine if two matrices can be multiplied, you need to 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 matrices can be multiplied.
Q: What is the resulting matrix when two matrices are multiplied?
A: The resulting matrix is a new matrix that has 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 obtained by multiplying the elements of the corresponding row of the first matrix with the elements of the corresponding column of the second matrix.
Q: How do I perform matrix multiplication?
A: To perform matrix multiplication, you need to follow these steps:
- Check if the number of columns in the first matrix is equal to the number of rows in the second matrix.
- If the condition is met, multiply the elements of the rows of the first matrix with the elements of the columns of 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.
Q: What are some common mistakes to avoid when performing matrix multiplication?
A: Some common mistakes to avoid when performing matrix multiplication include:
- Not checking if the number of columns in the first matrix is equal to the number of rows in the second matrix.
- Not following the correct order of matrix multiplication.
- Not performing the multiplication correctly.
Q: How do I use matrix multiplication in real-world applications?
A: Matrix multiplication has numerous applications in various fields such as physics, engineering, and computer science. Some of the applications of matrix multiplication include:
- Linear Transformations: Matrix multiplication is used to represent linear transformations in mathematics and physics.
- Computer Graphics: Matrix multiplication is used to perform transformations on 2D and 3D objects in computer graphics.
- 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 and matrix inversion.
Q: What are some advanced topics related to matrix multiplication?
A: Some advanced topics related to matrix multiplication include:
- Matrix Inversion: Matrix inversion is the process of finding the inverse of a matrix.
- Matrix Determinant: The matrix determinant is a scalar value that can be used to determine the invertibility of a matrix.
- Eigenvalues and Eigenvectors: Eigenvalues and eigenvectors are used to study the properties of matrices.
Conclusion
In this article, we have answered some of the most frequently asked questions about matrix multiplication. We have discussed the order of matrix multiplication, how to determine if two matrices can be multiplied, and how to perform matrix multiplication. We have also discussed some common mistakes to avoid when performing matrix multiplication and some advanced topics related to matrix multiplication.