Matrix Multiplication Calculate [5 4; 2 -1] * [-4; 3]

Introduction to Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra with vast applications across various fields, including computer graphics, data analysis, and physics. Understanding how to multiply matrices is crucial for solving complex problems and developing efficient algorithms. In this article, we will delve into the process of multiplying a 2x2 matrix by a 2x1 matrix, providing a step-by-step guide to solving the given problem: $\left[\begin{array}{cc}5 & 4 \\ 2 & -1\end{array}\right] \cdot\left[\begin{array}{c}-4 \\ 3\end{array}\right]$.

At its core, matrix multiplication involves combining the elements of two matrices in a specific way to produce a new matrix. The dimensions of the matrices involved play a crucial role in determining whether multiplication is possible and what the dimensions of the resulting matrix will be. Specifically, for two matrices A and B to be multiplied (A * B), the number of columns in A must be equal to the number of rows in B. If A is an m x n matrix and B is an n x p matrix, then the resulting matrix C will be an m x p matrix. This foundational rule ensures that the operation is well-defined and that the resulting matrix has the correct dimensions. Let’s begin by understanding the basics of matrices and their dimensions, which is essential for grasping the multiplication process. A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The dimensions of a matrix are expressed as the number of rows by the number of columns. For example, a 2x2 matrix has two rows and two columns, while a 2x1 matrix has two rows and one column. Before diving into the multiplication process, it's crucial to verify that the dimensions of the matrices are compatible for multiplication. In our case, we are multiplying a 2x2 matrix by a 2x1 matrix. The number of columns in the first matrix (2) is equal to the number of rows in the second matrix (2), so the multiplication is possible. The resulting matrix will have the dimensions of 2x1, which means it will have two rows and one column. This initial check is a critical step in matrix multiplication, as it prevents errors and ensures the operation is valid. The concept of matrix dimensions is also important for understanding the structure of the resulting matrix and interpreting its elements. Each element in the resulting matrix is calculated by taking the dot product of a row from the first matrix and a column from the second matrix. This process will be explained in detail in the next section. By carefully considering the dimensions of the matrices involved, we can accurately predict the size and shape of the final result, making the multiplication process more intuitive and less prone to errors.

Step-by-Step Solution

To multiply the matrices $\left[\begin{array}{cc}5 & 4 \\ 2 & -1\end{array}\right]$ and $\left[\begin{array}{c}-4 \\ 3\end{array}\right]$, we perform the following steps:

1. Identify the dimensions of the matrices: The first matrix is a 2x2 matrix, and the second matrix is a 2x1 matrix. Since the number of columns in the first matrix (2) equals the number of rows in the second matrix (2), we can proceed with the multiplication. The resulting matrix will be a 2x1 matrix.

2. Calculate the elements of the resulting matrix:

   • First element: Multiply the first row of the first matrix by the first (and only) column of the second matrix: (5 * -4) + (4 * 3) = -20 + 12 = -8.
   • Second element: Multiply the second row of the first matrix by the first column of the second matrix: (2 * -4) + (-1 * 3) = -8 - 3 = -11.

3. Write the resulting matrix: The resulting matrix is a 2x1 matrix with the calculated elements:

   $\left[\begin{array}{c}-8 \\ -11\end{array}\right]$

Thus, the product of the given matrices is $\left[\begin{array}{c}-8 \\ -11\end{array}\right]$. This step-by-step process ensures that each element in the resulting matrix is calculated correctly, reducing the chances of errors. In this particular case, we started by identifying the dimensions of the matrices, which is a crucial first step. By recognizing that we are multiplying a 2x2 matrix by a 2x1 matrix, we can confirm that the multiplication is valid and that the resulting matrix will be 2x1. This foundational understanding helps to frame the subsequent calculations and ensures that we are working within the correct mathematical context. Next, we focus on calculating the individual elements of the resulting matrix. The first element is obtained by taking the dot product of the first row of the 2x2 matrix and the first (and only) column of the 2x1 matrix. This involves multiplying corresponding elements and summing the results. In our case, this is (5 * -4) + (4 * 3), which simplifies to -20 + 12, yielding -8. Similarly, the second element is calculated by taking the dot product of the second row of the 2x2 matrix and the first column of the 2x1 matrix. This calculation is (2 * -4) + (-1 * 3), which simplifies to -8 - 3, giving us -11. These individual calculations are crucial for constructing the final resulting matrix. Finally, we assemble the calculated elements into the resulting 2x1 matrix. This matrix is represented as $\left[\begin{array}{c}-8 \\ -11\end{array}\right]$. This final step brings together all the individual calculations into a cohesive and meaningful result. The resulting matrix provides a concise representation of the transformation or operation that the original matrix multiplication represents. By following this systematic approach, we can confidently and accurately multiply matrices, even when dealing with more complex scenarios. The step-by-step method ensures clarity and reduces the potential for mistakes, making matrix multiplication a manageable and understandable process.

Understanding the Dot Product

The dot product is a fundamental operation in matrix multiplication. It is used to calculate the individual elements of the resulting matrix by combining the rows of the first matrix with the columns of the second matrix. To understand the dot product, consider two vectors or sequences of numbers. The dot product is the sum of the products of the corresponding entries. For example, if we have two vectors a = [a1, a2, ..., an] and b = [b1, b2, ..., bn], their dot product is calculated as:

a · b = a1 * b1 + a2 * b2 + ... + an * bn

In the context of matrix multiplication, the dot product is applied between the rows of the first matrix and the columns of the second matrix. Specifically, the element in the i-th row and j-th column of the resulting matrix is the dot product of the i-th row of the first matrix and the j-th column of the second matrix. Let's illustrate this with our example. We have the matrices:

$\left[\begin{array}{cc}5 & 4 \\ 2 & -1\end{array}\right]$ and $\left[\begin{array}{c}-4 \\ 3\end{array}\right]$

To find the first element of the resulting matrix (which will be in the first row), we take the dot product of the first row of the first matrix [5, 4] and the first column of the second matrix [-4, 3]. This is calculated as:

(5 * -4) + (4 * 3) = -20 + 12 = -8

Similarly, to find the second element of the resulting matrix (in the second row), we take the dot product of the second row of the first matrix [2, -1] and the first column of the second matrix [-4, 3]. This is calculated as:

(2 * -4) + (-1 * 3) = -8 - 3 = -11

Thus, the resulting matrix is:

$\left[\begin{array}{c}-8 \\ -11\end{array}\right]$

This process clearly demonstrates how the dot product is used to combine the elements of the matrices. The dot product provides a scalar value that represents the projection of one vector onto another, which is a crucial concept in linear algebra. Understanding the dot product not only clarifies the mechanics of matrix multiplication but also sheds light on its geometric interpretation. Each element in the resulting matrix is a measure of how much one vector (a row of the first matrix) aligns with another vector (a column of the second matrix). This alignment is quantified by the sum of the products of their corresponding components. The dot product's significance extends beyond mere computation; it reveals underlying relationships between vectors and matrices, making it an indispensable tool in various mathematical and scientific applications. By mastering the concept of the dot product, one can gain a deeper appreciation for the power and versatility of matrix multiplication. It is this operation that enables us to transform, combine, and analyze data in a wide range of contexts, from computer graphics to data analysis. In essence, the dot product is the fundamental building block upon which the entire process of matrix multiplication is built.

Applications of Matrix Multiplication

Matrix multiplication is not just a theoretical concept; it has numerous practical applications across various fields. Here are some key areas where matrix multiplication is essential:

1. Computer Graphics: In computer graphics, transformations such as scaling, rotation, and translation are represented using matrices. Multiplying these transformation matrices allows for complex transformations to be applied to objects efficiently. For instance, rotating a 3D object involves multiplying its vertices (represented as vectors) by a rotation matrix. By combining multiple transformations into a single matrix through multiplication, graphics engines can render complex scenes quickly and smoothly.
2. Linear Systems of Equations: Matrix multiplication is used to represent and solve systems of linear equations. A system of linear equations can be written in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix. Solving for X involves finding the inverse of matrix A and multiplying it by B. This is a fundamental application in engineering, economics, and many other fields.
3. Data Analysis and Machine Learning: In data analysis, matrices are used to represent datasets. Matrix multiplication is a core operation in many machine learning algorithms, including neural networks and principal component analysis (PCA). For example, in neural networks, matrix multiplication is used to compute the weighted sum of inputs at each layer. PCA uses matrix multiplication to reduce the dimensionality of data while preserving important information.
4. Physics: In physics, matrices are used to represent linear transformations, such as rotations and reflections. They are also used in quantum mechanics to represent operators and states. Matrix multiplication is used to calculate the evolution of quantum systems over time and to perform various other calculations.
5. Economics: Economists use matrices to model and analyze economic systems. For example, input-output models use matrices to represent the interdependencies between different sectors of an economy. Matrix multiplication is used to calculate the total output required to meet a given level of demand.

These applications highlight the versatility and importance of matrix multiplication. Its ability to efficiently perform complex calculations and represent relationships between data makes it an indispensable tool in many fields. The application in computer graphics is particularly noteworthy, where matrix multiplication enables the creation of realistic and interactive visual experiences. By representing transformations as matrices, complex operations can be performed quickly and efficiently, allowing for the rendering of intricate 3D scenes. Similarly, the use of matrix multiplication in solving linear systems is a cornerstone of many scientific and engineering disciplines. The ability to represent a set of equations in matrix form and solve for the unknowns provides a powerful framework for modeling and analyzing a wide range of systems. In the realm of data analysis and machine learning, matrix multiplication is a fundamental building block of many algorithms. From neural networks to PCA, the ability to perform matrix operations efficiently is crucial for processing large datasets and extracting meaningful insights. The applications in physics and economics further underscore the broad applicability of matrix multiplication. In physics, it is used to model quantum systems and perform calculations related to linear transformations. In economics, it provides a framework for analyzing interdependencies between different sectors of an economy. Overall, the widespread use of matrix multiplication across various fields demonstrates its power and importance as a mathematical tool. Its ability to represent complex relationships and perform efficient calculations makes it an essential skill for anyone working in science, engineering, or data analysis.

Conclusion

In this article, we have explored the process of multiplying the matrices $\left[\begin{array}{cc}5 & 4 \\ 2 & -1\end{array}\right]$ and $\left[\begin{array}{c}-4 \\ 3\end{array}\right]$. We have broken down the steps involved, emphasizing the importance of understanding matrix dimensions and the dot product. The resulting matrix is $\left[\begin{array}{c}-8 \\ -11\end{array}\right]$. This exercise illustrates the fundamental principles of matrix multiplication, a crucial operation in linear algebra with wide-ranging applications in various fields. From computer graphics to data analysis, matrix multiplication plays a vital role in solving complex problems and developing efficient algorithms. By mastering the techniques and concepts presented in this article, readers can gain a solid foundation in linear algebra and its applications. The significance of matrix multiplication extends beyond the mere calculation; it represents a powerful tool for transforming and manipulating data in a variety of contexts. The ability to efficiently perform matrix multiplication is essential for anyone working with large datasets, complex systems, or advanced mathematical models. The dot product, as a core component of matrix multiplication, provides a way to quantify the relationship between vectors, allowing for the measurement of alignment and projection. This concept is crucial for understanding the underlying geometric and algebraic properties of matrices. Furthermore, the applications of matrix multiplication highlight its versatility and practical importance. In computer graphics, it enables the creation of realistic 3D scenes by efficiently transforming objects in space. In data analysis and machine learning, it forms the backbone of many algorithms, allowing for the processing and analysis of vast amounts of data. In physics and economics, it provides a framework for modeling and analyzing complex systems. By understanding the fundamentals of matrix multiplication and its applications, one can unlock a wide range of possibilities in various fields. The ability to work with matrices and perform operations such as multiplication is a valuable skill that can lead to new insights and innovative solutions. This article serves as a starting point for further exploration of linear algebra and its applications, encouraging readers to delve deeper into this fascinating and powerful area of mathematics.