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Chapter 3: Problem 39

Let \(A\) be a square matrix. Prove that \(\left(A^{\mathrm{T}}\right)^{\mathrm{T}}=A\).

Short Answer

Expert verified
To prove that \(\left(A^{\mathrm{T}}\right)^{\mathrm{T}}=A\) for a square matrix A, we first find the transpose of A, denoted by \(A^T\), by interchanging its rows and columns. Then, we find the transpose of \(A^T\), denoted by \((A^T)^T\), by interchanging its rows and columns again. We observe that the elements of \((A^T)^T\) are the same as the elements of A, i.e., \(a_{ij}=b_{ji}\) for all i and j. Hence, we can conclude that \(\left(A^{\mathrm{T}}\right)^{\mathrm{T}}=A\).

Step by step solution

01

Understand the Transpose of a Matrix

The transpose of a given matrix is obtained by interchanging its rows and columns. In other words, if A is an m脳n matrix, then the transpose of A (denoted by A^T) is an n脳m matrix in which the rows of A become columns and the columns of A become rows. Formally, if A = [a_ij], then A^T = [b_ij], where b_ij = a_ji for all i and j.
02

Find the Transpose of the Transpose of A

Now, let's find the transpose of A^T, which is denoted by (A^T)^T. Recall that, to find the transpose of a matrix, we interchange its rows and columns. So, if A^T is a matrix with elements a_ji (the rows and columns of A are interchanged), then for (A^T)^T, we need to again interchange its rows and columns, which means the elements will be b_ji, where b_ji = a_ij for all i and j.
03

Compare (A^T)^T with A

Now that we have found the elements of (A^T)^T, let's compare them to the elements of A. We know that the (i, j) element of A is a_ij, and as we observed above, the (i, j) element of (A^T)^T is b_ji, which is equal to a_ij for all i and j. Since the elements of (A^T)^T are the same as the elements of A, we can conclude that (A^T)^T = A. Thus, we have proved that for a square matrix A, the transpose of its transpose is equal to the matrix itself: \(\left(A^{\mathrm{T}}\right)^{\mathrm{T}}=A\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Transposition Properties
Matrix transposition is a fundamental operation in linear algebra that reveals important properties of matrices. When we take the transpose of a matrix, denoted by the superscript T, we effectively interchange its rows and columns. Understanding the properties of matrix transposition aids in comprehending more complex linear transformations and operations.

One critical takeaway is that the transpose of a transpose brings us back to the original matrix, which is expressed in the equation \( (A^{\mathrm{T}})^{\mathrm{T}} = A \). This reflective property demonstrates that transposition is an involution. Moreover, the transpose of a product of matrices is equal to the product of their transposes in reverse order, or \( (AB)^{\mathrm{T}} = B^{\mathrm{T}}A^{\mathrm{T}} \). Transposition is also compatible with scalar multiplication, meaning that \( (kA)^{\mathrm{T}} = k(A^{\mathrm{T}}) \), where k is a scalar.
Linear Algebra
Linear algebra is a branch of mathematics that deals with vectors, vector spaces, linear mappings, and systems of linear equations. It is essential for various fields such as physics, computer science, engineering, and statistics. A matrix serves as a compact way to represent and operate on sets of linear equations.

In the context of our exercise, dealing with matrix transposition gives us insight into one of the many operations we can perform on matrices. Besides transposition, there are other concepts such as matrix multiplication, determinant calculation, inversion, and eigenvalue decomposition. Linear algebra not only provides the tools to manipulate and solve linear systems but also offers a language to describe geometric transformations, network flows, and optimization problems.
Proof Techniques in Mathematics
Proof techniques are the foundation on which mathematics is built. They provide a systematic way to establish the truthfulness of mathematical statements. The proof presented for the property \( (A^{\mathrm{T}})^{\mathrm{T}} = A \) is an example of a direct proof, where we directly show that the action of double transposition on matrix A yields the original matrix A.

Other common proof techniques include proof by contradiction, where we assume the opposite of what we want to prove and show that it leads to a contradiction, and proof by induction, useful for proving statements about integers. Understanding various proof techniques is invaluable for students as they allow for a deeper understanding of mathematical concepts and contribute to the development of logical reasoning skills.

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