Question

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{ccc}6 & -4 & -5 \\ -4 & 0 & 2 \\ -5 & 2 & -2\end{array}\right]\)

Short answer

Transpose: \( A^T = A \). Matrix is symmetric.

Step-by-step solution

Identify the Elements of Matrix A

The given matrix \( A \) is: \[ A = \begin{bmatrix} 6 & -4 & -5 \ -4 & 0 & 2 \ -5 & 2 & -2 \end{bmatrix} \]Each element of this matrix can be identified by its row \( i \) and column \( j \), such that the matrix element \( a_{ij} \) is 6 at position (1,1) and -2 at position (3,3), and so on.

Calculate the Transpose of Matrix A

The transpose of a matrix, denoted as \( A^T \), is obtained by swapping the rows and columns of \( A \). For the matrix \( A \):- The first row \([6, -4, -5]\) becomes the first column.- The second row \([-4, 0, 2]\) becomes the second column.- The third row \([-5, 2, -2]\) becomes the third column.Thus, the transpose \( A^T \) is:\[ A^T = \begin{bmatrix} 6 & -4 & -5 \ -4 & 0 & 2 \ -5 & 2 & -2 \end{bmatrix} \]

Determine the Type of Matrix

To determine the characteristics of \( A \), check various properties:- **Symmetric**: A matrix is symmetric if \( A = A^T \). Here, \( A^T = A \), so it is symmetric.- **Skew Symmetric**: A matrix is skew symmetric if \( A^T = -A \). This is not true for \( A \).- **Triangular**: A matrix is upper or lower triangular if all elements below or above, respectively, the main diagonal are zero. Neither condition is met.- **Diagonal**: A matrix is diagonal if all non-diagonal elements are zero. \( A \) is not diagonal since many off-diagonal elements are non-zero.Since \( A \) equals \( A^T \), it is classified as a symmetric matrix.

Key concepts

Symmetric Matrix

A symmetric matrix is a special kind of square matrix that mirrors itself across its main diagonal. In simple terms, for a matrix to be symmetric, the element at row \(i\) and column \(j\), denoted as \(a_{ij}\), must be equal to the element at row \(j\) and column \(i\), \(a_{ji}\). This essentially means that \(A = A^T\), where \(A^T\) is the transpose of matrix \(A\).
For instance, in the matrix \(A = \begin{bmatrix} 6 & -4 & -5 \ -4 & 0 & 2 \ -5 & 2 & -2 \end{bmatrix}\) the element in the first row, second column (-4) is the same as the one in the second row, first column (-4). This pattern holds throughout the matrix, confirming that \(A\) is symmetric.

• Saves computational effort by storing or processing only half of the matrix.
• Arises naturally in many mathematical and engineering problems, particularly when considering relationships or distances.

Matrix Properties

Matrices have various properties that help us classify and analyze them in terms of their structure and functions. Knowing these attributes makes it easier to solve complex mathematical problems, especially in linear algebra.
1. **Symmetric Matrices**: As mentioned earlier, symmetric matrices are equal to their transpose. They frequently appear in cases involving quadratic forms or optimization problems.2. **Skew Symmetric Matrices**: These matrices have zeros on the main diagonal, and for them, \(A^T = -A\). The elements are mirrored negatively across the main diagonal.3. **Triangular Matrices**: In triangular matrices, all entries above (upper triangular) or below (lower triangular) the main diagonal must be zero.4. **Diagonal Matrices**: These only have non-zero values on the main diagonal, making them a very simplified case of matrices.Understanding these different forms assists in recognizing the easiest paths to solutions, especially leveraging simplifications afforded by each type's unique characteristics.

Matrix Transposition Process

The matrix transposition process involves swapping the columns and rows of a given matrix. This is a straightforward yet powerful concept, impacting how matrices interact with arithmetic operations and solving systems of equations.
In particular, the transpose of a matrix \( A \), denoted \( A^T \), is formed by flipping the matrix over its diagonal. Hence, the element at \((i,j)\) in the original matrix becomes the element at \((j,i)\) in the transposed matrix.

• Reflects the structure and nature of data as often used in statistical analysis and geometry.
• Key operation in defining matrix products and solving linear systems.

For our matrix \(A = \begin{bmatrix} 6 & -4 & -5 \ -4 & 0 & 2 \ -5 & 2 & -2 \end{bmatrix}\), when transposed, it remains identical because it is symmetric by nature, demonstrating an important property of symmetric matrices where \( A^T = A \). The transposition, therefore, confirms the symmetry of matrix \(A\).