Question

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{ccc}4 & 0 & -2 \\ 0 & 2 & 3 \\ -2 & 3 & 6\end{array}\right]\)

Short answer

The transpose is \( A^T = A \), and the matrix is symmetric.

Step-by-step solution

Identify the Transpose

The transpose of a matrix is found by swapping its rows with its columns. In essence, the first row becomes the first column, the second row becomes the second column, and so on for matrix \( A \).The given matrix \( A \) is \[A = \begin{bmatrix} 4 & 0 & -2 \ 0 & 2 & 3 \ -2 & 3 & 6 \end{bmatrix}\]

Calculate the Transpose

Apply the transpose operation on matrix \( A \). Write the rows of matrix \( A \) as columns to find \( A^T \).So, the first row \([4, 0, -2]\) becomes the first column, the second row \([0, 2, 3]\) becomes the second column, and the third row \([-2, 3, 6]\) becomes the third column.Thus, \( A^T \) is given by:\[A^T = \begin{bmatrix} 4 & 0 & -2 \ 0 & 2 & 3 \ -2 & 3 & 6 \end{bmatrix}\]

Check Properties of the Matrix

Compare the original matrix \( A \) and its transpose \( A^T \). Since \( A \) is equal to \( A^T \), the matrix is symmetric.\( A \) is also neither upper nor lower triangular because it has non-zero elements both above and below its diagonal. It is also neither diagonal nor skew-symmetric.

Key concepts

Symmetric Matrix

A symmetric matrix is a special type of square matrix that is equal to its transpose. This means that when you take the transpose of a symmetric matrix, the result will be the same as the original matrix. For a matrix to be symmetric, it must satisfy the condition \( a_{ij} = a_{ji} \) for all indices \( i \) and \( j \). This means that the element in the i-th row and j-th column is the same as the element in the j-th row and i-th column.
In the context of our problem, the matrix \( A \) is given as \(A = \begin{bmatrix}4 & 0 & -2 \0 & 2 & 3 \-2 & 3 & 6\end{bmatrix}\) When we compute its transpose, we find that \( A^{T} \) is identical to \( A \). Therefore, \( A \) is classified as a symmetric matrix. Symmetric matrices are important in various fields such as physics, statistics, and mathematics as they possess unique properties and are often easier to work with in equations and algorithms.

Matrix Properties

Matrices have distinct properties that define their structure and influence their behavior in mathematical operations. Some important matrix properties relevant to our problem are:

• Symmetric: As mentioned, a symmetric matrix is equal to its transpose.
• Upper Triangular: A matrix is upper triangular if all elements below the diagonal are zero.
• Lower Triangular: Conversely, a matrix is lower triangular if all elements above the diagonal are zero.
• Diagonal: A diagonal matrix has all non-diagonal elements as zero.
• Skew-symmetric: A square matrix is skew-symmetric if its transpose is equal to the negative of the matrix \( (A^T = -A) \).

For matrix \( A \) in our exercise, it is not upper or lower triangular, as it has non-zero elements on both sides of the diagonal. It is also not diagonal because not all non-diagonal elements are zero. Lastly, it's not skew-symmetric since \( A^T eq -A \). Understanding these properties can aid in recognizing patterns and predicting results in matrix algebra.

Matrix Algebra

Matrix algebra involves various operations and rules for manipulating matrices, which are essential in solving mathematical problems. Key operations in matrix algebra include:

• Matrix Addition: Adding two matrices involves adding their corresponding elements.
• Scalar Multiplication: Each element of a matrix is multiplied by a scalar value.
• Matrix Multiplication: This operation involves the product of rows and columns between two matrices. Choosing the correct order is crucial since multiplication is not commutative.
• Transpose: Swapping the rows and columns of a matrix, as demonstrated by the transpose of \( A \).
• Determinant: A scalar value that provides important properties and information about a square matrix.

Matrix algebra is pivotal in a wide array of applications, from solving linear equations to transformations in computer graphics. The ability to apply these matrix operations and properties can significantly simplify complex problems in science and engineering.