What Is The Result Of The Scalar Multiplication \( 3 \cdot \begin{bmatrix} 1 & -4 & 5 & -7 \end{bmatrix} \)? Choose From The Following Options: A. \( \begin{bmatrix} 3 & -12 & 15 & -21 \end{bmatrix} \), B. \( \begin{bmatrix} 4 & -1 & 8 & -4 \end{bmatrix} \).

by ADMIN 259 views

In the realm of mathematics, particularly in linear algebra, scalar multiplication of matrices is a fundamental operation. It involves multiplying a matrix by a scalar (a real number), which results in a new matrix where each element is the product of the original element and the scalar. This concept is crucial for various applications, including computer graphics, data analysis, and solving systems of linear equations. In this article, we will delve into the specifics of scalar multiplication, focusing on the given example: 3⋅[1−45−7]{ 3 \cdot \begin{bmatrix} 1 & -4 & 5 & -7 \end{bmatrix} }. We will break down the process step-by-step, explain the underlying principles, and discuss the broader implications of scalar multiplication in matrix operations.

Understanding Scalar Multiplication

Scalar multiplication is a straightforward yet powerful operation. It allows us to scale a matrix, either increasing or decreasing the magnitude of its elements while preserving the matrix's dimensions. To perform scalar multiplication, you simply multiply each element of the matrix by the given scalar. Let's consider a general matrix A:

A=[a11a12a21a22]{ A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} }

If we multiply this matrix by a scalar k, the resulting matrix kA is:

kA=k⋅[a11a12a21a22]=[ka11ka12ka21ka22]{ kA = k \cdot \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} = \begin{bmatrix} ka_{11} & ka_{12} \\ ka_{21} & ka_{22} \end{bmatrix} }

This means that every element aij{ a_{ij} } in the original matrix A is multiplied by the scalar k to obtain the corresponding element in the resulting matrix kA. This simple rule forms the basis of scalar multiplication and is applicable to matrices of any size and dimension. Understanding this basic principle is essential before we tackle the specific problem at hand.

Step-by-Step Calculation of 3⋅[1−45−7]{ 3 \cdot \begin{bmatrix} 1 & -4 & 5 & -7 \end{bmatrix} }

Now, let's apply the concept of scalar multiplication to the given matrix and scalar. We are asked to find the product of the scalar 3 and the matrix [1−45−7]{ \begin{bmatrix} 1 & -4 & 5 & -7 \end{bmatrix} }. This is a 1x4 matrix (one row and four columns), and we will multiply each of its elements by the scalar 3. Here’s the step-by-step calculation:

  1. Multiply the first element (1) by 3: 3⋅1=3{ 3 \cdot 1 = 3 }
  2. Multiply the second element (-4) by 3: 3⋅(−4)=−12{ 3 \cdot (-4) = -12 }
  3. Multiply the third element (5) by 3: 3⋅5=15{ 3 \cdot 5 = 15 }
  4. Multiply the fourth element (-7) by 3: 3⋅(−7)=−21{ 3 \cdot (-7) = -21 }

Therefore, the resulting matrix is [3−1215−21]{ \begin{bmatrix} 3 & -12 & 15 & -21 \end{bmatrix} }. This process demonstrates the simplicity and directness of scalar multiplication. Each element is scaled by the scalar value, resulting in a new matrix with adjusted magnitudes.

Analyzing the Options

Given the calculated result, we can now compare it with the provided options:

  • A. [3−1215−21]{ \begin{bmatrix} 3 & -12 & 15 & -21 \end{bmatrix} }
  • B. [4−18−4]{ \begin{bmatrix} 4 & -1 & 8 & -4 \end{bmatrix} }

By comparing our calculated result [3−1215−21]{ \begin{bmatrix} 3 & -12 & 15 & -21 \end{bmatrix} } with the options, it is clear that option A matches our result. Option B, on the other hand, does not match the calculated values. This step highlights the importance of accurate calculation and comparison in matrix operations. Ensuring that each element is correctly multiplied and that the resulting matrix is accurately represented is crucial for reaching the correct solution.

The Correct Answer

Based on our calculations and analysis, the correct answer is:

A. [3−1215−21]{ \begin{bmatrix} 3 & -12 & 15 & -21 \end{bmatrix} }

This result confirms that scalar multiplication involves multiplying each element of the matrix by the scalar, resulting in a new matrix with scaled elements. The process is straightforward, but accuracy is key to avoiding errors. Scalar multiplication forms the basis for more complex matrix operations and is a fundamental concept in linear algebra. Mastering this operation is essential for understanding more advanced topics in the field.

The Significance of Scalar Multiplication in Linear Algebra

Scalar multiplication, while seemingly simple, plays a pivotal role in linear algebra and its applications. It is one of the fundamental operations that, along with matrix addition, defines the vector space structure. Understanding scalar multiplication is crucial for grasping more complex concepts such as linear transformations, eigenvalues, and eigenvectors. Scalar multiplication allows us to scale vectors and matrices, which is essential in many real-world applications.

Applications in Computer Graphics

In computer graphics, scalar multiplication is used extensively for scaling objects. For example, if you have a 3D model represented by a set of vertices (which can be represented as matrices), multiplying these vertices by a scalar allows you to resize the object. This is a basic operation in rendering and animation, enabling the creation of realistic visuals and dynamic scenes. By scaling objects, we can create the illusion of depth and distance, making scenes appear more realistic and immersive.

Applications in Data Analysis

In data analysis, scalar multiplication is used for normalization and standardization of data. Normalizing data involves scaling the values to a standard range, such as between 0 and 1. This is often necessary to ensure that different features in a dataset have comparable ranges, preventing features with larger values from dominating the analysis. Scalar multiplication is a key component in these processes, allowing data scientists to preprocess data effectively and improve the performance of machine learning models. Standardizing data helps in reducing bias and improving the accuracy of models by ensuring that all features contribute equally to the analysis.

Applications in Physics and Engineering

In physics and engineering, scalar multiplication is used to scale vectors representing forces, velocities, and other physical quantities. For instance, if you double the force applied to an object, you are essentially performing scalar multiplication. This is crucial in simulations and calculations where the magnitude of physical quantities needs to be adjusted. Understanding how to scale vectors and matrices is essential for solving problems in mechanics, electromagnetism, and other fields. The ability to manipulate these quantities through scalar multiplication allows engineers and physicists to model and predict the behavior of complex systems.

Linear Transformations

Scalar multiplication is also fundamental to the concept of linear transformations. A linear transformation is a function that maps vectors to vectors while preserving certain properties, such as linearity. Scalar multiplication is one of the operations that must be preserved in a linear transformation. This means that if you apply a linear transformation to a scalar multiple of a vector, the result is the same as applying the scalar multiple to the transformed vector. This property is crucial for understanding how linear transformations work and how they can be used to manipulate vectors and matrices. Linear transformations are used in a wide range of applications, from image processing to machine learning, making scalar multiplication an essential concept in these fields.

Conclusion: Mastering Scalar Multiplication

In conclusion, scalar multiplication is a fundamental operation in linear algebra with far-reaching implications. It is the process of multiplying a matrix by a scalar, where each element of the matrix is multiplied by the scalar value. This simple operation is crucial for a variety of applications, including computer graphics, data analysis, and physics. By mastering scalar multiplication, you lay the groundwork for understanding more advanced concepts in linear algebra and its applications. The example of 3⋅[1−45−7]{ 3 \cdot \begin{bmatrix} 1 & -4 & 5 & -7 \end{bmatrix} } demonstrates the straightforward nature of scalar multiplication and its importance in matrix operations. Understanding and practicing scalar multiplication is essential for anyone working with matrices and vectors, whether in academic or professional settings. The ability to accurately and efficiently perform this operation is a key skill for problem-solving and analysis in numerous fields.

By understanding the mechanics and applications of scalar multiplication, students and professionals can gain a deeper appreciation for the power and versatility of linear algebra. This foundational knowledge opens doors to more advanced topics and practical applications, making it an indispensable tool in the toolbox of mathematicians, scientists, and engineers.