Solving For Matrix A In A Matrix Equation

Introduction

In the realm of linear algebra, matrix equations stand as fundamental tools for representing and solving systems of linear equations. These equations involve matrices, which are rectangular arrays of numbers, and various operations defined on them. One such operation is matrix addition, where matrices of the same dimensions are added element-wise. This article delves into the intricacies of matrix equations, focusing on solving for an unknown matrix. Specifically, we aim to determine the value of matrix A in a given matrix equation. Understanding matrix operations and equation-solving techniques is crucial for various applications in mathematics, physics, engineering, and computer science. In the given matrix equation, we have A plus a known matrix equals another known matrix. Our goal is to isolate A and find its value by performing appropriate matrix operations. This involves understanding the properties of matrix addition and subtraction, ensuring that we manipulate the equation while maintaining its balance. The process of solving for A not only demonstrates the practical application of matrix algebra but also reinforces the fundamental concepts of linear transformations and vector spaces. By carefully examining the structure of the matrices and applying the rules of matrix arithmetic, we can systematically determine the elements of matrix A, providing a clear and concise solution to the problem. This approach highlights the power and elegance of matrix algebra in solving complex problems involving multiple variables and relationships. The ability to solve such equations is a cornerstone of many advanced mathematical and computational techniques, making this a valuable skill for students and professionals alike.

Problem Statement

We are presented with the matrix equation:

$A+\left[\begin{array}{cccc} 3 & 9 & -1 & -8 \\ 16 & -2 & 3 & 13 \end{array}\right]=\left[\begin{array}{cccc} 0 & -5 & 6 & 10 \\ 3 & 0 & -2 & 7 \end{array}\right]$

Our objective is to find the matrix A that satisfies this equation. To achieve this, we will employ the principles of matrix algebra, specifically the properties of matrix addition and subtraction. Matrix addition and subtraction are element-wise operations, meaning that corresponding elements in the matrices are added or subtracted. For two matrices to be added or subtracted, they must have the same dimensions. In this case, both the matrix being added to A and the resulting matrix are 2x4 matrices, so the operation is valid. The strategy to solve for A involves isolating it on one side of the equation. This is analogous to solving algebraic equations with scalars, where we perform inverse operations to isolate the variable. In the context of matrices, we will subtract the matrix added to A from both sides of the equation. This operation will effectively cancel out the matrix on the left side, leaving A isolated. The resulting matrix will then represent the solution for A. This process requires careful attention to the signs and positions of the elements within the matrices. By performing the subtraction accurately, we can determine the exact values of the elements in matrix A. This method not only provides the solution but also reinforces the understanding of how matrix operations work and how they can be used to solve complex problems. Understanding these principles is essential for further studies in linear algebra and its applications in various fields.

Solution

To find A, we subtract the matrix $\left[\begin{array}{cccc} 3 & 9 & -1 & -8 \\ 16 & -2 & 3 & 13 \end{array}\right]$ from both sides of the equation:

$A = \left[\begin{array}{cccc} 0 & -5 & 6 & 10 \\ 3 & 0 & -2 & 7 \end{array}\right] - \left[\begin{array}{cccc} 3 & 9 & -1 & -8 \\ 16 & -2 & 3 & 13 \end{array}\right]$

Now, we perform element-wise subtraction:

$A = \left[\begin{array}{cccc} 0-3 & -5-9 & 6-(-1) & 10-(-8) \\ 3-16 & 0-(-2) & -2-3 & 7-13 \end{array}\right]$

Simplifying the elements, we get:

$A = \left[\begin{array}{cccc} -3 & -14 & 7 & 18 \\ -13 & 2 & -5 & -6 \end{array}\right]$

Thus, the matrix A is $\left[\begin{array}{cccc} -3 & -14 & 7 & 18 \\ -13 & 2 & -5 & -6 \end{array}\right]$. This solution demonstrates the application of matrix subtraction, a fundamental operation in linear algebra. By subtracting the known matrix from both sides of the equation, we successfully isolated the unknown matrix A. The element-wise subtraction ensures that each corresponding entry in the matrices is properly accounted for, leading to the correct result. This process highlights the importance of understanding the rules of matrix arithmetic and their application in solving matrix equations. The solution for A now allows us to verify the original equation by adding it back to the matrix that was initially added to A. If the sum equals the matrix on the right-hand side of the original equation, then our solution is correct. This verification step is crucial in ensuring the accuracy of the result and reinforcing the understanding of matrix addition and subtraction. Moreover, this exercise provides a foundation for solving more complex matrix equations, including those involving matrix multiplication and inversion. The ability to manipulate matrices and solve equations involving them is a valuable skill in various fields, including engineering, computer science, and economics.

Conclusion

In conclusion, the value of matrix A in the given matrix equation is $\left[\begin{array}{cccc} -3 & -14 & 7 & 18 \\ -13 & 2 & -5 & -6 \end{array}\right]$. This was determined by applying the principles of matrix subtraction, where we subtracted the known matrix from both sides of the equation to isolate A. The process involved element-wise subtraction, ensuring that each corresponding entry in the matrices was properly handled. This solution showcases the fundamental techniques of matrix algebra, particularly the manipulation of matrices to solve equations. Matrix equations are a cornerstone of linear algebra, with applications spanning various fields. Understanding how to solve these equations is essential for tasks such as solving systems of linear equations, performing linear transformations, and analyzing data in various scientific and engineering contexts. The ability to isolate an unknown matrix and determine its elements is a crucial skill for anyone working with matrices. The steps taken in this solution, from setting up the equation to performing the subtraction and arriving at the final result, provide a clear and concise method for solving similar problems. Furthermore, the solution reinforces the importance of careful attention to detail and a thorough understanding of matrix operations. The concepts and techniques demonstrated here serve as a foundation for more advanced topics in linear algebra, such as matrix inversion, eigenvalues, and eigenvectors. By mastering these fundamental skills, students and professionals can effectively tackle complex problems involving matrices and linear systems. The result not only answers the specific question but also enhances the broader understanding of matrix algebra and its applications.