Question

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{cc}3 & 1 \\ -7 & 8\end{array}\right]\)

Short answer

The transpose is \( \begin{bmatrix} 3 & -7 \\ 1 & 8 \end{bmatrix} \) and it is neither symmetric nor skew-symmetric.

Step-by-step solution

Understanding the Matrix

The given matrix \( A \) is \[ \begin{bmatrix} 3 & 1 \ -7 & 8 \end{bmatrix} \] which is a 2x2 matrix with elements listed row-wise.

Find the Transpose

To find the transpose of a matrix \( A \), you flip the matrix over its diagonal. This means the rows of \( A \) become the columns of \( A^{T} \). So, \( A^{T} \) is \[ \begin{bmatrix} 3 & -7 \ 1 & 8 \end{bmatrix} \].

Analyze Matrix Properties

The original matrix \( A \) is neither upper triangular (as elements below the main diagonal are not all zero) nor lower triangular (as elements above the main diagonal are not all zero). It is not diagonal, as not all off-diagonal elements are zero. It is not symmetric because \( A eq A^{T} \). It is also not skew-symmetric because \( A eq -A^{T} \).

Key concepts

Matrix Properties

Matrix properties are essential in understanding the behavior of matrices. These properties can inform us about the nature of the matrix, and can significantly affect the calculations and operations we perform with them.

• Upper Triangular Matrix: A matrix is upper triangular if all the elements below the main diagonal are zero. This means you get a triangle (usually filled with zeros) in the bottom part of the matrix.
• Lower Triangular Matrix: Conversely, a matrix is lower triangular if all the elements above the main diagonal are zero. Here, the zeros form a triangle in the upper part of the matrix.
• Diagonal Matrix: This type has non-zero elements only on the main diagonal, with all other elements being zero.
• Symmetric Matrix: A matrix is symmetric if it equals its transpose. This means swapping rows with columns leaves the matrix unchanged.
• Skew-Symmetric Matrix: A skew-symmetric matrix is one where the transpose equals its negative. For example, if a matrix element `a` at position (i, j) is equal to the negative of the element at (j, i).

Understanding these properties can help simplify problems involving matrices, enabling us to approach complex calculations in a structured manner.
Knowing whether a matrix is triangular, diagonal, symmetric, or skew-symmetric helps in many mathematical contexts.

2x2 Matrix

A 2x2 matrix is one of the simplest forms in the realm of matrices, consisting of two rows and two columns. These matrices are often used in basic problems, offering a simple layout to understand complex concepts. Despite their simplicity, 2x2 matrices can still represent a plethora of properties and transformations. A given 2x2 matrix \[\begin{bmatrix} a & b \ c & d \\end{bmatrix}\]includes four key numbers (elements). Each position represents crucial information where:

• `a` and `d` are on the main diagonal.
• The other two elements, `b` and `c`, determine whether it's triangular or has other properties.

The operations involving 2x2 matrices form the foundation for more extensive matrix work. They are ideal for illustrating concepts like determinant, inverse, and simple transformations.

Matrix Symmetry

Matrix symmetry is an interesting property that greatly influences the matrix's characteristics. A symmetric matrix is equal to its transpose. This property is particularly useful in simplifying mathematical problems. For a matrix to be symmetric, the elements across the main diagonal must mirror each other. In mathematical terms, if a matrix is \[A = \begin{bmatrix} a & b \ c & d \\end{bmatrix}\]then it is symmetric if `a = a`, `b = c`, and `d = d`. This implies that the transposed matrix looks exactly the same as the original. For example, if we examine matrix \[\begin{bmatrix} 1 & 2 \ 2 & 1 \\end{bmatrix}\],we see it is symmetric because swapping rows and columns doesn’t alter its form.Many real-world phenomena are captured with symmetric matrices because of their stability traits. They appear frequently in statistics, physics, and engineering.

Matrix Types

Understanding different matrix types helps in various fields of calculation and analysis. Although there are many, several common matrix types frequently appear in mathematical problems.

• Square Matrix: A matrix is square if it has the same number of rows and columns. Examples include a 2x2 or 3x3 matrix.
• Rectangular Matrix: This matrix has a different number of rows and columns. For instance, it could be 2x3 or 3x2.
• Identity Matrix: A specific type of diagonal matrix where all elements on the main diagonal are 1, and all other elements are 0. An identity matrix acts as a multiplicative identity in matrix multiplication.
• Zero Matrix: All elements are zero, often used as an additive identity in matrix operations.
• Orthogonal Matrix: Two matrices are orthogonal if their product with the transpose equals the identity matrix, playing a pivotal role in transformations.

Different matrix types serve various purposes, from solving equations to modeling transformations or representing data in compact forms. Recognizing these types can help you choose the best approach for matrix methods and analysis.