Evaluate Matrix Expression 1/2M + 2N Step By Step

Introduction

In the realm of linear algebra, evaluating expressions involving matrices is a fundamental skill. This article delves into a step-by-step evaluation of the matrix expression ${ \frac{1}{2}M + 2N }$, where ${ M = \begin{bmatrix} 1 & 2 \\ 0 & -1 \end{bmatrix} }$ and ${ N = \begin{bmatrix} 3 & -1 \\ 2 & 4 \end{bmatrix} }$. We will explore the processes of scalar multiplication and matrix addition, providing a clear and comprehensive guide suitable for students and enthusiasts alike. Mastering these operations is crucial for understanding more complex concepts in mathematics, physics, and computer science. Matrix operations form the backbone of various applications, from solving systems of linear equations to transforming data in machine learning algorithms. Therefore, a solid grasp of these fundamentals is essential for anyone venturing into these fields. This discussion will not only present the solution but also elucidate the underlying principles, ensuring a thorough understanding of the methods involved. Through detailed explanations and intermediate steps, readers will gain confidence in handling similar matrix expressions. By the end of this article, you should be able to confidently tackle matrix expressions involving scalar multiplication and addition, paving the way for more advanced topics in linear algebra.

Scalar Multiplication

Before we can add the matrices, we need to perform scalar multiplication. This involves multiplying each element of a matrix by a scalar value. In our expression, we have two scalar multiplications to perform: ${ \frac{1}{2}M }$ and ${ 2N }$. Let's start with ${ \frac{1}{2}M }$. To compute ${ \frac{1}{2}M }$, we multiply each element of matrix M by the scalar ${ \frac{1}{2} }$. Given that ${ M = \begin{bmatrix} 1 & 2 \\ 0 & -1 \end{bmatrix} }$, the calculation proceeds as follows:

${ \frac{1}{2}M = \frac{1}{2} \begin{bmatrix} 1 & 2 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} \frac{1}{2} \times 1 & \frac{1}{2} \times 2 \\ \frac{1}{2} \times 0 & \frac{1}{2} \times -1 \end{bmatrix} = \begin{bmatrix} \frac{1}{2} & 1 \\ 0 & -\frac{1}{2} \end{bmatrix} }$

Next, we compute ${ 2N }$. We multiply each element of matrix N by the scalar 2. Given that ${ N = \begin{bmatrix} 3 & -1 \\ 2 & 4 \end{bmatrix} }$, the calculation is:

${ 2N = 2 \begin{bmatrix} 3 & -1 \\ 2 & 4 \end{bmatrix} = \begin{bmatrix} 2 \times 3 & 2 \times -1 \\ 2 \times 2 & 2 \times 4 \end{bmatrix} = \begin{bmatrix} 6 & -2 \\ 4 & 8 \end{bmatrix} }$

Scalar multiplication is a straightforward operation, but it is a crucial step in many matrix calculations. By performing scalar multiplication, we are essentially scaling the matrix, which can have significant implications in various applications. For instance, in computer graphics, scaling transformations are used to resize objects. Similarly, in image processing, scalar multiplication can be used to adjust the brightness of an image. Therefore, understanding and mastering scalar multiplication is essential for anyone working with matrices. This foundational knowledge allows for more complex operations and analyses, underlining the importance of this step in the overall evaluation of matrix expressions. Correct execution of scalar multiplication is paramount, as any error here will propagate through subsequent operations, leading to an incorrect final result. By meticulously performing each multiplication, we ensure the accuracy of our calculations and build a solid foundation for understanding more advanced matrix manipulations.

Matrix Addition

Now that we have performed the scalar multiplications, we can proceed with matrix addition. Matrix addition is performed by adding corresponding elements of the matrices. For this operation to be valid, the matrices must have the same dimensions. In our case, both ${ \frac{1}{2}M }$ and ${ 2N }$ are 2x2 matrices, so we can add them. We have:

${ \frac{1}{2}M = \begin{bmatrix} \frac{1}{2} & 1 \\ 0 & -\frac{1}{2} \end{bmatrix} \quad \text{and} \quad 2N = \begin{bmatrix} 6 & -2 \\ 4 & 8 \end{bmatrix} }$

To add these matrices, we add the corresponding elements:

${ \frac{1}{2}M + 2N = \begin{bmatrix} \frac{1}{2} & 1 \\ 0 & -\frac{1}{2} \end{bmatrix} + \begin{bmatrix} 6 & -2 \\ 4 & 8 \end{bmatrix} = \begin{bmatrix} \frac{1}{2} + 6 & 1 + (-2) \\ 0 + 4 & -\frac{1}{2} + 8 \end{bmatrix} }$

Now, we perform the additions element-wise:

${ \begin{bmatrix} \frac{1}{2} + 6 & 1 + (-2) \\ 0 + 4 & -\frac{1}{2} + 8 \end{bmatrix} = \begin{bmatrix} \frac{1}{2} + \frac{12}{2} & 1 - 2 \\ 4 & -\frac{1}{2} + \frac{16}{2} \end{bmatrix} = \begin{bmatrix} \frac{13}{2} & -1 \\ 4 & \frac{15}{2} \end{bmatrix} }$

Thus, the result of the matrix addition is:

${ \frac{1}{2}M + 2N = \begin{bmatrix} \frac{13}{2} & -1 \\ 4 & \frac{15}{2} \end{bmatrix} }$

Matrix addition is a fundamental operation in linear algebra, allowing us to combine matrices in a meaningful way. The ability to add matrices is essential in various applications, such as solving systems of linear equations, performing transformations in computer graphics, and analyzing data in statistics. The principle of adding corresponding elements ensures that the matrices are combined in a consistent and logical manner. Moreover, the requirement that matrices must have the same dimensions for addition highlights the importance of dimensional consistency in matrix operations. This principle ensures that we are adding elements that correspond to the same positions within the matrices, maintaining the structural integrity of the data. Understanding matrix addition is crucial for progressing to more complex matrix operations and applications, reinforcing its significance in the broader context of linear algebra.

Final Result

After performing the scalar multiplication and matrix addition, we have successfully evaluated the expression ${ \frac{1}{2}M + 2N }$. The final result is:

${ \frac{1}{2}M + 2N = \begin{bmatrix} \frac{13}{2} & -1 \\ 4 & \frac{15}{2} \end{bmatrix} }$

This resulting matrix is a 2x2 matrix, as expected, given that both M and N are 2x2 matrices. The elements of the resulting matrix are obtained through careful application of scalar multiplication and matrix addition rules. Each element represents the sum of the scaled corresponding elements from the original matrices. This result underscores the importance of meticulous calculations in linear algebra, as each step builds upon the previous one, and any error can propagate through the entire process. By breaking down the expression into smaller, manageable steps, we have demonstrated a systematic approach to evaluating matrix expressions. This approach not only yields the correct result but also fosters a deeper understanding of the underlying principles. The final matrix represents a linear combination of the original matrices, scaled by the constants ${ \frac{1}{2} }$ and 2, respectively. This linear combination is a fundamental concept in linear algebra, with applications ranging from solving systems of equations to representing transformations in vector spaces. The ability to accurately compute such linear combinations is essential for advanced topics in mathematics, physics, and engineering.

Conclusion

In this article, we have thoroughly evaluated the matrix expression ${ \frac{1}{2}M + 2N }$, where ${ M = \begin{bmatrix} 1 & 2 \\ 0 & -1 \end{bmatrix} }$ and ${ N = \begin{bmatrix} 3 & -1 \\ 2 & 4 \end{bmatrix} }$. We began by performing scalar multiplication, multiplying matrix M by ${ \frac{1}{2} }$ and matrix N by 2. Then, we added the resulting matrices, element by element, to obtain the final matrix. The step-by-step approach used in this evaluation highlights the importance of understanding the fundamental operations of scalar multiplication and matrix addition. These operations are not only essential for solving matrix expressions but also form the building blocks for more advanced concepts in linear algebra. The final result, ${ \begin{bmatrix} \frac{13}{2} & -1 \\ 4 & \frac{15}{2} \end{bmatrix} }$, demonstrates the linear combination of matrices, which is a core concept in various fields, including mathematics, physics, computer science, and engineering. Mastering these matrix operations is crucial for anyone working with data, transformations, and systems of equations. The principles of linear algebra provide a powerful toolkit for solving complex problems, and a solid understanding of matrix operations is the key to unlocking this potential. By presenting a clear and detailed solution, this article aims to empower readers with the skills and knowledge needed to confidently tackle similar matrix expressions. The ability to evaluate matrix expressions is a valuable asset in many domains, and the techniques discussed here provide a solid foundation for further exploration and application of linear algebra concepts.