Given Matrix A = [[6, 2], [5, 6]] And A³ + A = 0, Determine The Correct Relationship Between X And Y.
In the fascinating realm of linear algebra, matrices hold a central position, serving as fundamental building blocks for various mathematical and computational applications. Matrix equations, which involve matrices as variables, offer a powerful tool for modeling and solving a wide range of problems in physics, engineering, economics, and computer science. In this article, we delve into the intricacies of a specific matrix equation, A³ + A = 0, where A represents a 2x2 matrix. Our objective is to explore the properties of this equation and determine the conditions that must be satisfied by the elements of matrix A for the equation to hold true. We will embark on a step-by-step analysis, leveraging key concepts from linear algebra, such as matrix multiplication, matrix addition, and the concept of the zero matrix. By meticulously examining the equation, we will uncover the relationships between the elements of matrix A and derive the conditions that govern its solutions. This exploration will not only enhance our understanding of matrix equations but also provide valuable insights into the broader field of linear algebra.
Problem Statement
Let's consider the matrix A, defined as:
A = egin{pmatrix} 6 & 2 \ 5 & 6
\end{pmatrix}
We are given the matrix equation:
A³ + A = 0
where 0 represents the zero matrix.
Our task is to determine the conditions that must be satisfied by the elements of matrix A for the given equation to hold true. In other words, we aim to find the relationships between the elements of A that ensure the equation A³ + A = 0 is satisfied.
Step-by-Step Solution
1. Calculate A²
To begin our analysis, we first need to compute A², which is the product of matrix A with itself:
A² = A * A
A² = egin{pmatrix} 6 & 2 \ 5 & 6
\end{pmatrix} * egin{pmatrix} 6 & 2 \ 5 & 6
\end{pmatrix}
Performing the matrix multiplication, we get:
A² = egin{pmatrix} (66 + 25) & (62 + 26) \ (56 + 65) & (52 + 66)
\end{pmatrix}
A² = egin{pmatrix} 46 & 24 \ 60 & 46
\end{pmatrix}
2. Calculate A³
Next, we need to calculate A³, which is the product of A² and A:
A³ = A² * A
A³ = egin{pmatrix} 46 & 24 \ 60 & 46
\end{pmatrix} * egin{pmatrix} 6 & 2 \ 5 & 6
\end{pmatrix}
Performing the matrix multiplication, we get:
A³ = egin{pmatrix} (466 + 245) & (462 + 246) \ (606 + 465) & (602 + 466)
\end{pmatrix}
A³ = egin{pmatrix} 396 & 236 \ 590 & 396
\end{pmatrix}
3. Substitute into the Equation A³ + A = 0
Now, we substitute the calculated values of A³ and A into the given equation:
A³ + A = egin{pmatrix} 396 & 236 \ 590 & 396
\end{pmatrix} + egin{pmatrix} 6 & 2 \ 5 & 6
\end{pmatrix}
A³ + A = egin{pmatrix} (396 + 6) & (236 + 2) \ (590 + 5) & (396 + 6)
\end{pmatrix}
A³ + A = egin{pmatrix} 402 & 238 \ 595 & 402
\end{pmatrix}
4. Analyze the Result
We are given that A³ + A = 0, where 0 represents the zero matrix:
0 = egin{pmatrix} 0 & 0 \ 0 & 0
\end{pmatrix}
However, our calculation shows that:
A³ + A = egin{pmatrix} 402 & 238 \ 595 & 402
\end{pmatrix}
This result clearly indicates that A³ + A ≠ 0 for the given matrix A. Therefore, there seems to be an inconsistency in the problem statement or the given matrix A. It is possible that there was a typo or an error in the original problem.
Alternative Approach and Generalization
Let's consider a general 2x2 matrix A of the form:
A = egin{pmatrix} x & y \ z & w
\end{pmatrix}
where x, y, z, and w are real numbers. We will now explore the conditions under which A³ + A = 0 holds true for this general matrix.
1. Calculate A² for the General Matrix
A² = A * A
A² = egin{pmatrix} x & y \ z & w
\end{pmatrix} * egin{pmatrix} x & y \ z & w
\end{pmatrix}
A² = egin{pmatrix} (x² + yz) & (xy + yw) \ (zx + wz) & (zy + w²)
\end{pmatrix}
2. Calculate A³ for the General Matrix
A³ = A² * A
A³ = egin{pmatrix} (x² + yz) & (xy + yw) \ (zx + wz) & (zy + w²)
\end{pmatrix} * egin{pmatrix} x & y \ z & w
\end{pmatrix}
A³ = egin{pmatrix} ((x² + yz)x + (xy + yw)z) & ((x² + yz)y + (xy + yw)w) \ ((zx + wz)x + (zy + w²)z) & ((zx + wz)y + (zy + w²)w)
\end{pmatrix}
Simplifying, we get:
A³ = egin{pmatrix} (x³ + 2xyz + yzw) & (x²y + xyw + yw² + y²z) \ (x²z + xzw + zw² + z²y) & (xyz + 2zyw + w³)
\end{pmatrix}
3. Substitute into the Equation A³ + A = 0
Now, we substitute A³ and A into the equation A³ + A = 0:
egin{pmatrix} (x³ + 2xyz + yzw) & (x²y + xyw + yw² + y²z) \ (x²z + xzw + zw² + z²y) & (xyz + 2zyw + w³)
\end{pmatrix} + egin{pmatrix} x & y \ z & w
\end{pmatrix} = egin{pmatrix} 0 & 0 \ 0 & 0
\end{pmatrix}
This gives us the following system of equations:
- x³ + 2xyz + yzw + x = 0
- x²y + xyw + yw² + y²z + y = 0
- x²z + xzw + zw² + z²y + z = 0
- xyz + 2zyw + w³ + w = 0
4. Analyze the System of Equations
Analyzing this system of equations is quite complex. However, we can gain some insights by considering specific cases.
Case 1: A is a Scalar Multiple of the Identity Matrix
Let's assume A = kI, where k is a scalar and I is the identity matrix:
A = egin{pmatrix} k & 0 \ 0 & k
\end{pmatrix}
In this case, the equation A³ + A = 0 becomes:
(kI)³ + kI = 0
k³I + kI = 0
(k³ + k)I = 0
This implies that k³ + k = 0, which simplifies to k(k² + 1) = 0. The solutions for k are k = 0 and k = ±i (where i is the imaginary unit).
Case 2: Other Solutions
Finding other solutions to the system of equations requires more advanced techniques, such as considering the characteristic equation of matrix A. The characteristic equation is given by:
|A - λI| = 0
where λ represents the eigenvalues of A. For the equation A³ + A = 0 to hold, the eigenvalues of A must satisfy the equation λ³ + λ = 0, which means λ(λ² + 1) = 0. The solutions for λ are λ = 0 and λ = ±i.
Conclusion
In this article, we embarked on a detailed exploration of the matrix equation A³ + A = 0. We first attempted to solve the equation for a specific 2x2 matrix A, but our calculations revealed an inconsistency in the problem statement. We then shifted our focus to a more general approach, considering a general 2x2 matrix and deriving a system of equations that must be satisfied for the equation to hold true. While analyzing the general case proved to be complex, we gained valuable insights by considering specific scenarios, such as when A is a scalar multiple of the identity matrix. We also touched upon the concept of eigenvalues and their role in determining the solutions of matrix equations.
This exploration has highlighted the importance of careful analysis and attention to detail when dealing with matrix equations. It has also demonstrated the power of leveraging key concepts from linear algebra, such as matrix multiplication, matrix addition, and eigenvalues, to solve complex problems. While we were not able to find a specific solution for the given matrix A, our analysis has provided a deeper understanding of the properties of the equation A³ + A = 0 and the conditions that must be met for it to hold true.
Final Answer
Based on our initial calculations, the given matrix A does not satisfy the equation A³ + A = 0. Therefore, there is likely an error in the problem statement or the given matrix. However, the general analysis provides a framework for solving similar matrix equations by considering the relationships between the elements of the matrix and the eigenvalues.
Regarding the options provided:
- A) xy = 1
- B) x = y
- C) xy + 1 = 0
- D) xy + 2 = 0
Without a specific valid matrix A that satisfies A³ + A = 0, we cannot definitively choose one of these options. The general analysis suggests that the relationship between the elements of A would be more complex than these simple equations.