Question

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{ccc}0 & 1 & -2 \\ -1 & 0 & 4 \\ 2 & -4 & 0\end{array}\right]\)

Short answer

A is skew-symmetric with transpose \(A^T = \begin{bmatrix} 0 & -1 & 2 \\ 1 & 0 & -4 \\ -2 & 4 & 0 \end{bmatrix}\).

Step-by-step solution

Understand Transpose of a Matrix

To find the transpose of a matrix, we swap rows with columns. For example, the first row of the matrix becomes the first column of the transpose, the second row becomes the second column, and so on.

Apply Transpose to Matrix A

Given matrix A:\[\begin{bmatrix} 0 & 1 & -2 \ -1 & 0 & 4 \ 2 & -4 & 0 \end{bmatrix}\]The first row \([0, 1, -2]\) becomes the first column, the second row \([-1, 0, 4]\) becomes the second column, and the third row \([2, -4, 0]\) becomes the third column. Hence, the transpose is:\[A^T = \begin{bmatrix} 0 & -1 & 2 \ 1 & 0 & -4 \ -2 & 4 & 0 \end{bmatrix}\]

Determine if Matrix A has Any Special Properties

Matrix A is a square matrix (same number of rows and columns), so we check for the following properties:- **Upper or Lower Triangular:** A triangular matrix has all zeros either above or below the main diagonal, respectively. Matrix A has non-zero values both above and below the diagonal, so it is not upper or lower triangular.- **Diagonal:** A diagonal matrix has non-zero elements only on the main diagonal. Since A has zeros on the diagonal, but some non-zero entries elsewhere, it is not diagonal.- **Symmetric:** A symmetric matrix is equal to its transpose, i.e., \(A = A^T\). Since \(A eq A^T\), it is not symmetric.- **Skew-Symmetric:** A skew-symmetric matrix satisfies \(A = -A^T\) and has zeros on the main diagonal. The matrix A satisfies \(A = -A^T\) as \[-A = \begin{bmatrix} 0 & -1 & 2 \ 1 & 0 & -4 \ -2 & 4 & 0 \end{bmatrix}\] matches \(A^T\). Hence, A is skew-symmetric.

Key concepts

Square Matrix

A square matrix is one where the number of rows is equal to the number of columns. This uniformity is crucial as it allows for certain mathematical operations and properties to be explored.

• Square matrices appear frequently in linear algebra and are used to solve systems of linear equations.
• They are also pivotal in understanding concepts like eigenvalues and eigenvectors.
• A square matrix of order 3, for instance, features three rows and three columns, making it perfectly symmetrical in terms of size, not necessarily in terms of its elements.

Square matrices can further be categorized into special types based on the positions and values of their elements, such as triangular, diagonal, or symmetric matrices. These categories help in simplifying matrix operations and in understanding deeper properties about the matrices.

Skew-Symmetric Matrix

A skew-symmetric matrix comes with a unique set of properties that make it interesting. A matrix \(A\) is skew-symmetric if it satisfies the condition \(A = -A^T\). This means that when you transpose the matrix and then change the sign of all its elements, you get back the original matrix.

• In a skew-symmetric matrix, the diagonal elements are always zero. This is because each diagonal element must be equal to its own negative, which is only possible if they are zero.
• The off-diagonal elements are symmetric with respect to sign; that is, if one element \(a_{ij}\) is positive, then \(a_{ji}\) must be negative of that element.
• Examples and applications include tensors in physics and certain types of transformations which conserve orientation.

Understanding skew-symmetric matrices is vital for grasping concepts in advanced linear algebra, as they provide insights into antisymmetric properties in multidimensional spaces.

Triangular Matrix

Triangular matrices are another fascinating category of square matrices. These matrices are defined by their non-zero elements residing exclusively on and either above or below the main diagonal. We have two types:

• Upper Triangular Matrix: All non-zero elements are on or above the main diagonal, while everything below it is zero.
• Lower Triangular Matrix: All non-zero elements are on or below the main diagonal, while everything above it is zero.

These matrices simplify many matrix computations. For instance, they make solving systems of linear equations easier, especially when using techniques like forward or backward substitution.
Triangular matrices often arise in decomposition processes, such as LU decomposition, and are vital tools in numerical analysis and computer algorithms. Their structure provides a framework for efficiently understanding and solving complex mathematical problems.