Symmetric Matrices: A Deep Dive Into Matrix Operations

Oct 14, 2025 by Blender 55 views

Hey guys! Let's dive into the fascinating world of symmetric matrices and explore some interesting properties. This article will break down the concept of symmetric matrices, their characteristics, and how they interact with other matrices, focusing on the problem presented. We'll examine some key statements about matrix operations and determine their validity. Ready to get started? Let's go!

Understanding Symmetric Matrices

First off, what exactly is a symmetric matrix? Simply put, a symmetric matrix is a square matrix that is equal to its transpose. In other words, if you flip the matrix over its main diagonal (from the top left to the bottom right), nothing changes. This seemingly simple property has some profound implications and is crucial for understanding the problem at hand. We denote the transpose of a matrix A as A^T. So, a matrix B is symmetric if B = B^T. This means the element at the i-th row and j-th column (b_ij) is equal to the element at the j-th row and i-th column (b_ji). Easy peasy, right?

This symmetry isn't just a mathematical quirk; it pops up all over the place! Symmetric matrices appear in various fields, like physics (describing systems with energy conservation), engineering (analyzing structures and circuits), and even in data science (covariance matrices in statistics). It's a fundamental concept, so it's essential to get a good grasp of it. The key takeaway here is that a symmetric matrix has a special structure that simplifies many calculations and analyses. This structure is what we are going to use to evaluate the given statements. Now, let's look at the given problem.

Analyzing Matrix Operations: Unveiling Symmetry

Alright, let's get into the core of the problem. We're given two n x n matrices, A and B, and B is a symmetric matrix (B = B^T). Our mission is to evaluate the symmetry of the results of some matrix operations. We need to determine whether the resulting matrices from the given operations are also symmetric. The statements we are given are:

(I) ABBA is symmetric.
(II) A + A^T + B is symmetric.
(III) ABA is symmetric.

To determine the validity of these statements, we need to understand how matrix operations like multiplication and addition affect the symmetry of a matrix. Remember the properties of the transpose operator are key. The transpose of a product of matrices is the product of the transposes in reverse order ( (XY)^T = Y^TX^T ). Also, the transpose of a sum is the sum of the transposes ( (X + Y)^T = X^T + Y^T). Let's now carefully look at each statement one by one.

Dissecting the Statements: Step-by-Step Analysis

Statement (I): ABBA is symmetric?

Let's analyze the first statement, which claims that ABBA is symmetric. To check this, we need to find the transpose of ABBA and see if it equals ABBA. Using the transpose properties, we get:

(ABBA)^T = A^TB^TB^TA^T.

Since B is symmetric, B = B^T. So, we can replace B^T with B: A^TBB^TA^T = A^TBBA^T. However, without knowing anything about A, we cannot simplify this further to ABBA. It's not necessarily true that A^TBBA^T equals ABBA. The original statement might be false.

Now, let's explore if we can find a counterexample. If we could construct matrices A and B that satisfy the conditions but for which ABBA is not symmetric, we'd have a proof that statement (I) is false. Let's try that, assuming it may not be true and searching for a counterexample.

Statement (II): A + A^T + B is symmetric?

Let's see if the second statement (A + A^T + B) is symmetric. To verify this, we compute the transpose:

(A + A^T + B)^T = A^T + (A^T)^T + B^T = A^T + A + B

This is because (A^T)^T = A, and B^T = B (since B is symmetric). Now, by commutative property of addition, we can rearrange the terms. A^T + A + B = A + A^T + B. This means (A + A^T + B)^T = A + A^T + B. Thus, the statement is true. The second statement is guaranteed to be symmetric. This is an important clue to solving the problem.

Statement (III): ABA is symmetric?

For the third statement, let's analyze ABA. We'll take the transpose:

(ABA)^T = A^TB^TA^T.

Since B is symmetric, we can replace B^T with B, resulting in A^TBA^T. The matrix ABA is symmetric only if A^TBA^T = ABA. This is generally not the case unless A has specific properties that cause this equality. This is because the position of A in the operation has changed. The transpose of A and A has different positions, which doesn't guarantee the original result. So, the third statement is not always true.

Determining the Correct Answer

Let's wrap it up and find the correct option. We have determined the following about the statements:

(I) ABBA is not always symmetric.
(II) A + A^T + B is symmetric.
(III) ABA is not always symmetric.

Based on these findings, we can conclude that only statement (II) is true. Therefore, the correct answer is the option that indicates that only statement (II) is true.

So, the answer would be b) apenas (II) é verdadeira.

Conclusion: Symmetry Unveiled

In this article, we've explored the fascinating realm of symmetric matrices and their behavior under matrix operations. We've seen how the symmetry of a matrix impacts the symmetry of the resulting matrix after operations such as multiplication and addition. Understanding these concepts is essential for a deeper understanding of linear algebra and its applications. Keep practicing and exploring, guys! You got this! Remember the main points: how the transpose works, the properties of the matrices, and how each operation can change those properties. Good luck!