Finding Matrix A Given Matrix Equation 3A - 4I = 5[[1, -2], [0, -1]]
Introduction
In the realm of linear algebra, matrix equations are fundamental. They allow us to express and solve complex relationships between matrices. This article delves into a specific problem involving finding an unknown matrix, denoted as A, given a matrix equation. The equation involves the unit matrix I, scalar multiplication, and matrix subtraction and equality. We will explore the steps to solve this equation and the underlying concepts that make it possible. This article will help you understand matrix operations and their applications in solving algebraic problems. Understanding these concepts is crucial for various fields, including computer graphics, data analysis, and engineering. This comprehensive guide will walk you through the process step-by-step, ensuring a clear understanding of each operation involved. Let's begin by understanding the importance of matrix algebra and its applications in real-world scenarios.
Matrix algebra is not just a theoretical concept; it's a powerful tool used in various practical applications. From solving systems of linear equations to transforming objects in computer graphics, matrices play a crucial role. In this article, we focus on a specific problem, but the techniques and concepts discussed are widely applicable. For instance, in engineering, matrices are used to analyze structural systems and solve circuit networks. In computer science, they are essential for image processing and machine learning algorithms. By mastering the basics of matrix algebra, you can unlock a wide range of problem-solving capabilities.
Problem Statement
The problem we aim to solve is as follows:
Given that I is the unit matrix of order 2 × 2 and 3A - 4I = 5[1 -2; 0 -1], find the matrix A. This problem is a classic example of a matrix equation, where we need to isolate the unknown matrix A. To do this, we'll use properties of matrix arithmetic, including scalar multiplication, matrix addition, and the concept of the identity matrix. The solution involves a series of steps, each building upon the previous one, to finally arrive at the matrix A. Understanding how to manipulate matrix equations is a fundamental skill in linear algebra, and this problem serves as an excellent example to illustrate the process.
The unit matrix, often denoted by I, is a square matrix with ones on the main diagonal and zeros elsewhere. For a 2 × 2 matrix, the unit matrix is [[1, 0]; [0, 1]]. It plays a role similar to the number 1 in scalar arithmetic; that is, when a matrix is multiplied by the unit matrix, the original matrix remains unchanged. This property is crucial in solving matrix equations, as it allows us to isolate unknown matrices. Scalar multiplication involves multiplying a matrix by a scalar (a number), where each element of the matrix is multiplied by the scalar. Matrix addition and subtraction are performed element-wise, meaning corresponding elements in the matrices are added or subtracted. These basic operations form the foundation for solving more complex matrix equations.
Solution Steps
1. Define the Unit Matrix I
The unit matrix I, also known as the identity matrix, is a square matrix with ones on the main diagonal and zeros elsewhere. For a 2 × 2 matrix, I is defined as:
I = [[1, 0],
[0, 1]]
The unit matrix is a fundamental concept in linear algebra. It acts as the multiplicative identity, meaning that when any matrix is multiplied by the unit matrix, the original matrix remains unchanged. This property is crucial for solving matrix equations, as it allows us to isolate and manipulate matrices without altering their fundamental values. Understanding the properties of the unit matrix is essential for performing various matrix operations, such as finding inverses and solving systems of linear equations. The unit matrix serves as a building block for more complex matrix operations and is a cornerstone of linear algebra.
The identity matrix plays a similar role in matrix multiplication as the number 1 does in scalar multiplication. Just as multiplying a number by 1 leaves the number unchanged, multiplying a matrix by the identity matrix leaves the matrix unchanged. This property makes the identity matrix invaluable for solving matrix equations, as we can use it to isolate and solve for unknown matrices. For instance, if we have an equation AX = B, where A and B are known matrices and X is the unknown matrix, we can multiply both sides by the inverse of A (if it exists) to isolate X. The identity matrix helps us in this process by ensuring that A⁻¹A = I, which simplifies the equation to IX = A⁻¹B, or X = A⁻¹B.
2. Substitute I into the Equation
We are given the equation:
3A - 4I = 5 * [[1, -2],
[0, -1]]
Substitute the value of I:
3A - 4 * [[1, 0],
[0, 1]] = 5 * [[1, -2],
[0, -1]]
This substitution is a crucial step in solving the matrix equation. By replacing the symbolic representation of the unit matrix I with its actual matrix form, we can begin to perform the necessary arithmetic operations. This substitution allows us to transform the abstract equation into a concrete one, where we can apply the rules of scalar multiplication and matrix addition to isolate the unknown matrix A. Without this step, the equation would remain in a symbolic form, making it difficult to proceed towards a solution. The substitution of the unit matrix is a fundamental technique in solving matrix equations and is essential for progressing towards the final answer.
Once we substitute the value of I, we can then proceed with the scalar multiplication. This involves multiplying the scalar quantities with their respective matrices. Scalar multiplication is a straightforward process where each element of the matrix is multiplied by the scalar. For example, 4 * [[1, 0], [0, 1]] becomes [[4, 0], [0, 4]]. Similarly, 5 * [[1, -2], [0, -1]] becomes [[5, -10], [0, -5]]. These operations simplify the equation further and bring us closer to isolating the unknown matrix A. After performing scalar multiplication, we can then move on to matrix addition and subtraction to solve for A.
3. Perform Scalar Multiplication
Multiply the matrices by their respective scalars:
3A - [[4, 0],
[0, 4]] = [[5, -10],
[0, -5]]
Scalar multiplication is a fundamental operation in matrix algebra. It involves multiplying a matrix by a scalar (a number), and the result is a new matrix where each element is the product of the original element and the scalar. This operation is crucial for scaling matrices and is a key component in solving matrix equations. In our case, we multiply the unit matrix I by 4 and the matrix on the right-hand side by 5. This step simplifies the equation and allows us to isolate the term involving the unknown matrix A. Understanding scalar multiplication is essential for manipulating matrices and solving linear algebra problems.
After performing scalar multiplication, the equation becomes more manageable. We have effectively distributed the scalars across the matrices, resulting in simpler matrix expressions. This step prepares the equation for further manipulation, specifically matrix addition. By performing scalar multiplication, we have transformed the original equation into a form where we can now add matrices and isolate the term containing the unknown matrix A. This methodical approach of breaking down the problem into smaller, manageable steps is a key strategy in solving complex matrix equations. The next step involves adding the resulting matrices to further simplify the equation and move closer to the solution.
4. Isolate 3A
Add [[4, 0], [0, 4]] to both sides of the equation:
3A = [[5, -10],
[0, -5]] + [[4, 0],
[0, 4]]
Isolating the term 3A is a critical step in solving for the matrix A. To do this, we use the principle of adding the same matrix to both sides of the equation, which maintains the equality. In this case, we add the matrix [[4, 0], [0, 4]] to both sides, which effectively cancels out the -4I term on the left-hand side. This step is analogous to isolating a variable in a scalar equation, where we perform the same operation on both sides to maintain balance. By isolating 3A, we are one step closer to finding the matrix A itself. This process demonstrates the importance of understanding matrix operations and their properties in solving matrix equations.
Adding the same matrix to both sides of an equation is a fundamental principle in matrix algebra, just as it is in scalar algebra. This principle allows us to manipulate equations while preserving their validity. In this case, adding [[4, 0], [0, 4]] to both sides of the equation effectively moves the -4I term to the right-hand side, but as a positive term. This is a crucial step in isolating the unknown matrix A. By understanding and applying this principle, we can systematically solve for unknown matrices in complex equations. The next step involves performing the matrix addition on the right-hand side to further simplify the equation and bring us closer to the final solution.
5. Perform Matrix Addition
Add the matrices on the right side:
3A = [[5 + 4, -10 + 0],
[0 + 0, -5 + 4]]
3A = [[9, -10],
[0, -1]]
Matrix addition is performed element-wise, meaning we add corresponding elements in the matrices. This operation is a fundamental part of matrix algebra and is essential for solving matrix equations. In this step, we add the matrices on the right-hand side of the equation to simplify it further. The result is a new matrix where each element is the sum of the corresponding elements in the original matrices. This process reduces the complexity of the equation and brings us closer to isolating the unknown matrix A. Understanding matrix addition is crucial for manipulating matrices and solving linear algebra problems.
Element-wise addition is a straightforward process that involves adding the elements in the same positions in the matrices. For instance, the element in the first row and first column of the resulting matrix is the sum of the elements in the first row and first column of the matrices being added. This process is repeated for each element in the matrices. The result is a new matrix that represents the sum of the original matrices. This operation is a building block for more complex matrix operations and is essential for solving various problems in linear algebra. After performing matrix addition, we have a simplified equation that allows us to solve for the unknown matrix A in the next step.
6. Solve for A
Divide both sides by 3 (multiply by 1/3):
A = (1/3) * [[9, -10],
[0, -1]]
A = [[9/3, -10/3],
[0/3, -1/3]]
A = [[3, -10/3],
[0, -1/3]]
Solving for A involves isolating the matrix by performing scalar multiplication. Since we have 3A, we multiply both sides of the equation by 1/3 to obtain A. This step is analogous to dividing both sides of a scalar equation by a constant to solve for a variable. Scalar multiplication involves multiplying each element of the matrix by the scalar (1/3 in this case). This operation gives us the final solution for the matrix A. Understanding how to solve for a matrix in an equation is a fundamental skill in linear algebra, and this step demonstrates the process clearly.
Multiplying a matrix by a scalar involves distributing the scalar across each element of the matrix. This is a straightforward operation but is crucial for solving matrix equations. In this case, we multiply each element of the matrix [[9, -10], [0, -1]] by 1/3. This results in a new matrix where each element is the product of the original element and 1/3. This final step gives us the solution for the matrix A. By understanding and applying scalar multiplication, we can effectively solve for unknown matrices in various algebraic problems. The result is the matrix A, which satisfies the given equation.
Final Answer
Therefore, the matrix A is:
A = [[3, -10/3],
[0, -1/3]]
The final answer is the matrix A, which we have found by systematically solving the given matrix equation. This solution demonstrates the application of various matrix operations, including scalar multiplication, matrix addition, and the use of the unit matrix. The process involved isolating the unknown matrix A by performing the same operations on both sides of the equation, ensuring that the equality is maintained. This methodical approach is crucial for solving matrix equations and is a fundamental skill in linear algebra. The final answer provides the specific matrix that satisfies the original equation, completing the problem.
This final matrix A is the unique solution to the given equation. We can verify this solution by substituting A back into the original equation and checking if the equality holds. This verification step is always a good practice to ensure that the solution is correct. The solution process involved several key steps, including understanding the unit matrix, performing scalar multiplication, and using matrix addition. These operations are fundamental to linear algebra and are used extensively in various fields, including engineering, computer science, and mathematics. The successful solution of this problem demonstrates a strong understanding of matrix operations and their application in solving algebraic problems.
Conclusion
In conclusion, we have successfully found the matrix A by solving the given matrix equation. The solution involved understanding and applying the properties of the unit matrix, scalar multiplication, and matrix addition. This problem highlights the importance of matrix algebra in solving complex equations and provides a clear example of how to manipulate matrices to isolate unknowns. The techniques used in this solution are applicable to a wide range of matrix problems and are essential for anyone studying linear algebra. Mastering these concepts is crucial for various fields that rely on matrix operations, such as engineering, computer science, and data analysis.
The process of solving this matrix equation demonstrates a systematic approach that can be applied to other similar problems. The key is to understand the properties of matrix operations and to apply them in a logical sequence. By breaking down the problem into smaller, manageable steps, we can effectively solve for unknown matrices. This approach not only helps in solving specific problems but also enhances the understanding of matrix algebra in general. The concepts and techniques discussed in this article provide a solid foundation for further exploration of linear algebra and its applications in various fields. Matrix algebra is a powerful tool, and mastering its fundamentals opens up a wide range of problem-solving capabilities.
