Question

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{cccc}4 & -7 & -4 & -9 \\ -9 & 6 & 3 & -9\end{array}\right]\)

Short answer

The transpose \(A^T\) is a 4x2 matrix. It's non-square, so it cannot be triangular, diagonal, symmetric, or skew-symmetric.

Step-by-step solution

Identify the Matrix Type

The given matrix is \(A = \begin{bmatrix} 4 & -7 & -4 & -9 \ -9 & 6 & 3 & -9 \end{bmatrix}\). This matrix has 2 rows and 4 columns, making it a 2x4 matrix. Since the matrix is not square, it cannot be diagonal, upper triangular, or lower triangular. Additionally, non-square matrices cannot be symmetric or skew-symmetric.

Compute the Transpose

The transpose of a matrix \(A\), denoted by \(A^T\), is obtained by converting its rows into columns. Thus, the transpose of the given matrix \(A\) is: \[ A^T = \begin{bmatrix} 4 & -9 \ -7 & 6 \ -4 & 3 \ -9 & -9 \end{bmatrix} \]This results in a 4x2 matrix.

Determine Properties of Transpose

After finding the transpose \(A^T\), observe that it is a 4x2 matrix (not square), so it cannot be symmetric, skew-symmetric, or diagonal. Non-square matrices also cannot be classified as upper or lower triangular.

Key concepts

Matrix Types

Matrices come in various types, and understanding these is fundamental in matrix algebra. The basic type is a **square matrix**, where the number of rows is equal to the number of columns. This kind of matrix can further be classified into subtypes like **diagonal matrices**, where all non-diagonal elements are zero; **upper triangular matrices**, which have all zeros below the main diagonal; and **lower triangular matrices**, which have all zeros above the main diagonal. On the other hand, **non-square matrices** are those in which the number of rows does not equal the number of columns, making them quite different from their square counterparts.
Non-square matrices cannot be diagonal, upper, or lower triangular by definition, due to the imbalance in the number of rows and columns. Hence, they possess different properties and are often used in specific applications like graph theory and statistics. Understanding these types is crucial for classifying and applying matrices effectively in various mathematical problems.

Matrix Properties

Matrices exhibit a variety of properties that are crucial to their classification and application in algebra. These properties include *symmetry* and characteristics related to their composition.
An important property is **symmetry** in square matrices, where a matrix is symmetric if it equals its transpose; mathematically speaking, a matrix \(A\) is symmetric if \(A = A^T\). Another fundamental property is being **skew-symmetric**, where a matrix satisfies \(A = -A^T\).
For non-square matrices, due to the mismatch in row and column numbers, the notions of symmetry and skew-symmetry do not apply. However, understanding these concepts is critical when analyzing square matrices because they determine matrix behavior under operations such as transposition and multiplication.

Non-Square Matrices

Non-square matrices are matrices where the number of rows does not equal the number of columns. This means they diverge from the more commonly discussed square matrices, leading to unique characteristics and limitations in their properties. The matrix \(\begin{bmatrix} 4 & -7 & -4 & -9 \ -9 & 6 & 3 & -9 \end{bmatrix}\)is an example of a **2x4 matrix**, which illustrates this concept as it has 2 rows and 4 columns.
Non-square matrices cannot be **symmetric** or **skew-symmetric** since these properties are applicable only to matrices where rows equal columns. Additionally, non-square matrices cannot achieve **triangular forms**, such as upper or lower triangular, because these forms require that all elements below or above the diagonal (which only exists in square matrices) are zeros. This distinct structure makes non-square matrices suitable for specialized tasks, such as representing augmented matrices in systems of linear equations or in machine learning for handling data sets of differing features and samples.

Matrix Symmetry

Matrix symmetry is an interesting concept in matrix algebra, primarily applicable to square matrices. A matrix is called **symmetric** if it is equal to its own transpose; mathematically, for a matrix \(A\), it must fulfill the condition \(A = A^T\). When this occurs, the elements above and below the diagonal are mirrored, which simplifies many mathematical operations.
In contrast, a matrix is called **skew-symmetric** when it satisfies \(A = -A^T\). This means for any element \(a_{ij}\), \(a_{ij} = -a_{ji}\), and all diagonal elements are zero.
These concepts do not apply to **non-square matrices** like the one in our exercise because they lack a clear diagonal which facilitates these symmetry properties. However, for square matrices, symmetry and skew-symmetry can significantly affect how a matrix behaves under operations like multiplication, making these properties essential for deeper mathematical explorations.