Question

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{cc}13 & -3 \\ -3 & 1\end{array}\right]\)

Short answer

\(A\) is symmetric, and \(A^T = A\).

Step-by-step solution

Understanding Transposition

To find the transpose of a matrix, interchange its rows and columns. This means that every element at position \((i, j)\) in the original matrix is moved to position \((j, i)\) in the transposed matrix.

Transpose the Matrix

Given the matrix \(A = \begin{bmatrix} 13 & -3 \ -3 & 1 \end{bmatrix}\), the transpose \(A^T\) is found by swapping rows with columns. So:\[ A^T = \begin{bmatrix} 13 & -3 \ -3 & 1 \end{bmatrix} \].

Analyze the Transposed Matrix

The transposed matrix \(A^T\) is the same as the original matrix \(A\), meaning \(A\) is equal to \(A^T\). This shows that matrix \(A\) is symmetric, since a symmetric matrix is equal to its transpose.

Key concepts

Symmetric Matrix

A symmetric matrix is a type of square matrix that remains unchanged when transposed. Essentially, if a matrix is symmetric, it will be equal to its transpose, denoted as \( A^T \). Symmetry in matrices is a fascinating concept because it implies that the elements mirrored across the main diagonal (where row index \( i \) equals column index \( j \)) are identical.

For example, consider a symmetric matrix \( A \):

• If the element in the first row and second column is -3, then the element in the second row and first column is also -3.
• The main diagonal elements remain unchanged during transposition.

Symmetric matrices often arise in various domains, such as physics and computer science, especially in contexts involving coordinate transformations and linear equations. They simplify many matrix computations, making them an attractive choice where possible.

Matrix Operations

Matrix operations refer to the various procedures you can perform on matrices, such as addition, subtraction, multiplication, and transposition.

Let's delve into transposition, a fundamental matrix operation. Transposing a matrix involves flipping it over its diagonal, turning the matrix's rows into its columns. This operation is denoted by the notation \( A^T \) for a matrix \( A \). The important properties of transposition include:

• A matrix \( A \) and its transpose \( A^T \) have the same main diagonal elements.
• The transpose of a transpose returns the original matrix: \((A^T)^T = A\).
• A symmetric matrix remains the same after transposition: \( A = A^T \).

Understanding these operations can greatly aid in recognizing patterns and properties within more complex matrix computations, like solving linear equations or transforming geometric spaces.

Triangular Matrices

Triangular matrices come in two varieties: upper triangular and lower triangular. These matrices are special because they simplify matrix arithmetic significantly.

An upper triangular matrix is one where all the elements below the main diagonal are zero. Conversely, a lower triangular matrix has all elements above the main diagonal set to zero. These kinds of matrices can take the form:

• An upper triangular matrix might look like \( \begin{bmatrix} x & y \ 0 & z \end{bmatrix} \).
• A lower triangular matrix might appear as \( \begin{bmatrix} x & 0 \ w & z \end{bmatrix} \).

One of the key benefits of triangular matrices is their ability to simplify solving linear systems using processes like Gaussian elimination. They make it easier to perform complex computations because of their zero elements concentrated either above or below the main diagonal. Knowing how to identify and utilize triangular matrices is crucial for efficient computation and understanding matrix structure.