Question

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{llll}-9 & 8 & 2 & -7\end{array}\right]\)

Short answer

Transpose: \( A^T = \begin{bmatrix} -9 \\ 8 \\ 2 \\ -7 \end{bmatrix}\); not square, so no special properties.

Step-by-step solution

Understanding the Matrix

The given matrix \( A \) is \( \left[\begin{array}{llll}-9 & 8 & 2 & -7\end{array}\right] \). This matrix is a 1x4 matrix, which means it has only one row and four columns. When we talk about triangular, diagonal, symmetric, or skew symmetric matrices, these properties typically apply to square matrices. Since our matrix is 1x4, it does not meet the criteria for these properties.

Finding the Transpose of Matrix A

The transpose of a matrix \( A \), denoted \( A^T \), is obtained by converting its rows into columns. The original matrix is \( 1 \times 4 \), and its transpose will be \( 4 \times 1 \). Therefore, \( A^T \) will be:\[A^T = \begin{bmatrix}-9 \8 \2 \-7\end{bmatrix}\]

Final Verification and Notes

Since \( A \) is a 1x4 matrix, it cannot be classified as upper triangular, lower triangular, diagonal, symmetric, or skew-symmetric, as these properties apply to square matrices (matrices with the same number of rows and columns).

Key concepts

Matrix Properties

Matrices are mathematical objects representing organized numbers in rows and columns. These structures are crucial for many areas of mathematics and engineering. Different matrices have unique properties:

• **Triangular Matrices**: Matrices are classified as upper or lower triangular if they have only zeroes either below or above the diagonal, respectively.
• **Diagonal Matrices**: Where non-zero elements only exist along the diagonal from the top-left to the bottom-right.
• **Symmetric Matrices**: These matrices are equal to their transpose, meaning they look the same if flipped over their main diagonal.
• **Skew-Symmetric Matrices**: Here, the transpose gives the negative of the original matrix. The elements along the diagonal must be zero.

These properties typically pertain to square matrices, where the number of rows matches the number of columns.

Square Matrices

A square matrix is a matrix with the same number of rows and columns. These are essential because they allow for the definition of operations such as determinants and inverses. In mathematics, many properties and theorems apply specifically to square matrices.

• **Determinant**: A scalar value that can be calculated from its elements, helping in analyses such as solving linear systems.
• **Invertibility**: An invertible or non-singular matrix has an inverse, while a singular matrix does not.

Square matrices are often used to represent point transformations in linear algebra, making them incredibly useful in computer graphics and physics.

Transposition

The transpose of a matrix is a new matrix achieved by swapping rows with columns. This operation is noted as \(A^T\). For a matrix \(A\) with dimensions \(m \times n\), its transpose \(A^T\) will have dimensions \(n \times m\).

Transposition is a straightforward yet fundamental operation. For example, if matrix \(A\) is \(\begin{bmatrix}a & b & c \d & e & fe\end{bmatrix}\), the transpose \(A^T\) will be \(\begin{bmatrix} a & d \ b & e \ c & f\end{bmatrix}\).

• The transpose helps in defining symmetric and skew-symmetric matrices.
• It is also used in forming orthogonal matrices and understanding matrix symmetries.

Matrix Classification

Matrix classification segments matrices into different types based on their properties and shapes. Key classifications involve whether they are rectangular or square, and further properties such as being diagonal, triangular, symmetric or skew-symmetric.

Some broad classifications include:

• **Rectangular Matrices**: Have different numbers of rows and columns, such as 1x4 or 3x2.
• **Square Matrices**: Equal rows and columns, like 3x3 or 4x4.
• **Symmetric and Skew-Symmetric Matrices**: Defined by behaviors under transposition.
• **Identity Matrices**: A special kind of diagonal matrix with ones on the diagonal and zeroes elsewhere.

Understanding these classifications aids in knowing which matrix operations or properties can apply, showcasing the versatility and utility of matrices in mathematics and applied sciences.