Question

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{llll}-7 & -8 & 2 & -3\end{array}\right]\)

Short answer

Transpose of matrix A is a column vector.

Step-by-step solution

Identify the Matrix Type

The given matrix \( A \) is \([-7, -8, 2, -3]\). Since it is a 1x4 matrix, it does not have any non-zero elements above or below a main diagonal; therefore, it is not triangular. A diagonal matrix would require it to be square with all off-diagonal elements zero, so it is not diagonal either. It can't be symmetric or skew symmetric because those properties require a square matrix. Therefore, \( A \) is none of these types.

Find the Transpose of Matrix A

The transpose of a matrix \( A \), denoted as \( A^T \), is obtained by switching the rows and columns. For the given 1x4 matrix \( A = [-7, -8, 2, -3] \), the transpose \( A^T \) will be a 4x1 matrix:\[A^T = \begin{bmatrix}-7 \ -8 \ 2 \ -3 \end{bmatrix}\]

Key concepts

Matrix Types

Matrices can be found in several types based on their properties and dimensions. Understanding these types helps identify how matrices function or interact with mathematical operations. Here's a look at some common matrix types:

• Square Matrix: A matrix with the same number of rows and columns. For example, a 2x2 or 3x3 matrix is a square matrix.
• Rectangular Matrix: Such matrices have differing numbers of rows and columns, like our 1x4 matrix in the exercise.
• Diagonal Matrix: Only non-zero elements lie on the main diagonal, and it has to be a square matrix.
• Triangular Matrix: Either upper or lower triangular; elements are zero either above or below the diagonal. Again, it is typically square.
• Symmetric Matrix: If the matrix equals its transpose, such as when \( A = A^T \).
• Skew-Symmetric Matrix: When a matrix equals the negative of its transpose, \( A = -A^T \), it has to be square as well.

Our given matrix is a 1x4 row matrix, meaning it doesn't fit into any of the square-related categories like diagonal, symmetric, or triangular.

Transpose of a Matrix

The transpose of a matrix involves a simple yet vital operation in linear algebra. To transpose a matrix, one must switch its rows with its columns.
For example, if you have a matrix \( A \) with dimensions \( m \times n \), its transpose, denoted \( A^T \), will have dimensions \( n \times m \). Specifically for a row matrix like the one in our exercise, converting a 1x4 into a 4x1 column matrix results in: \[A^T = \begin{bmatrix}-7 \-8 \2 \-3 \end{bmatrix}\]
The transposition operation can modify the dimension structure of matrices and is critical in operations such as dot products and certain matrix equations.

Matrix Properties

Matrix properties encompass various characteristics that can determine how matrices interact in mathematical computations. Here are some essential properties:

• Commutative Property: Generally, matrix addition is commutative, but matrix multiplication is not. \( A + B = B + A \) holds, but \( AB eq BA \).
• Associative Property: Both matrix addition and multiplication are associative. This means \( (A + B) + C = A + (B + C) \) and \( (AB)C = A(BC) \).
• Distributive Property: Matrix multiplication distributes over addition, such as \( A(B + C) = AB + AC \).
• Identity Matrix: An identity matrix works like 1 in multiplication; \( AI = A \) and \( IA = A \).
• Zero Matrix: A zero matrix has all elements as zero, acting like zero in addition: \( A + 0 = A \).

Understanding and identifying these properties allows for more efficient computing and provides insight into matrix behavior during calculations.