Question

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{cc}-7 & 4 \\ 4 & -6\end{array}\right]\)

Short answer

The transpose is \(A^T = \left[\begin{array}{cc}-7 & 4 \\ 4 & -6\end{array}\right]\), and the matrix is symmetric.

Step-by-step solution

Identify Transpose

To find the transpose of matrix \(A\), we need to interchange its rows and columns. The given matrix \(A\) is \(\left[\begin{array}{cc}-7 & 4 \ 4 & -6\end{array}\right]\).

Perform Transpose Operation

Interchange the rows and columns of matrix \(A\). This means the element in the first row, first column of the original matrix will stay in the first row, first column, the element in the first row, second column will move to the second row, first column, and so on. Thus, the transpose \(A^T\) will be: \[A^{T} = \left[\begin{array}{cc}-7 & 4 \ 4 & -6\end{array}\right]\]

Analyze Matrix Properties

Inspect the given matrix and its transpose: - **Upper Triangular**: All elements below the main diagonal are zero. This is not true for \(A\) as elements below the main diagonal are \(4\) and \(-6\). - **Lower Triangular**: All elements above the main diagonal are zero. This is also not true for \(A\). - **Diagonal**: Only main diagonal has non-zero elements. This is not true for \(A\). - **Symmetric**: The matrix is equal to its transpose, \(A = A^T\). This is true for \(A\). - **Skew Symmetric**: The matrix is equal to the negative of its transpose, \(A = -A^T\). This is false for \(A\).

Conclusion

After analyzing matrix \(A\) and its transpose, we determine that the matrix \(A\) is **symmetric** because it is equal to its transpose.

Key concepts

symmetric matrix

A symmetric matrix is a special type of matrix that remains unchanged when it is transposed. In simpler terms, the matrix looks the same even if we swap its rows with columns. This means the element in the first row and first column is equal to the element in the first column and first row, and similarly for other elements across the main diagonal. For a matrix to be symmetric, it must meet the condition that for all indices, the element at position (i, j) is equal to the element at position (j, i). In the example given, the matrix \[\begin{array}{cc}-7 & 4 \4 & -6\end{array}\]is symmetric because swapping the values across the diagonal does not alter the matrix.Symmetric matrices often arise in the context of problems involving quadratic forms or in physics, where they reflect problems with certain symmetrical properties. Recognizing a symmetric matrix is essential because it often simplifies computations, especially in linear algebraic applications.

matrix properties

Understanding the properties of a matrix can provide insights into its behavior and potential applications in mathematics and science. Here are some key properties to consider:

• Upper Triangular: These matrices have all zero elements below the main diagonal, which runs from the top left to the bottom right. They are not symmetric unless they are diagonal as well.
• Lower Triangular: These are matrices where all elements above the main diagonal are zero. Like upper triangular matrices, they can be symmetric only if they are also diagonal.
• Diagonal: A diagonal matrix has non-zero elements only on its main diagonal and zero elsewhere. It is always symmetric.
• Symmetric: A matrix that is equal to its transpose is called symmetric. This means it's mirrored across its main diagonal.
• Skew Symmetric: This property indicates that a matrix is equal to the negative of its transpose, meaning each element satisfies \(a_{ij} = -a_{ji}\).

Knowing these matrix properties simplifies many algebraic operations and aids in recognizing patterns or predicting the outcomes of matrix manipulations.

transpose of a matrix

The transpose of a matrix is a fundamental operation in linear algebra where a matrix's rows are flipped into columns, and vice versa. Think of it as rotating the matrix by 90 degrees. If you have a matrix \( A \) with elements \[\begin{array}{cc}a_{11} & a_{12} \a_{21} & a_{22}\end{array}\]the transpose \( A^T \) will be\[\begin{array}{cc}a_{11} & a_{21} \a_{12} & a_{22}\end{array}\]Finding the transpose is simple:

• Swap the items across the main diagonal.
• A row becomes a column corresponding to its initial position.
• Effectively, this changes the matrix's orientation but not its properties concerning determinant or rank.

This concept is crucial in performing mathematical operations like solving linear equations or simplifying complex expressions. Transposing helps verify if a matrix is symmetric or to derive new properties when combined with other operations. Understanding how to transpose a matrix ensures accurate algebraic computations in various fields.