Question

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1\end{array}\right]\)

Short answer

Transposed matrix is the same: Symmetric, diagonal, upper and lower triangular. Not skew symmetric.

Step-by-step solution

Understand Matrix Transposition

The given matrix is \[ A = \begin{bmatrix} 1 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & -1 \end{bmatrix} \]Transposing a matrix involves swapping its rows with columns. For example, the first row becomes the first column, the second row becomes the second column, and so on.

Transpose the Matrix

Transpose the matrix \( A \) by turning rows into columns:\[ A^T = \begin{bmatrix} 1 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & -1 \end{bmatrix} \]Since the given matrix is already square and has diagonal entries, the transpose is equal to the original matrix.

Identify Matrix Properties

Analyze the original matrix:- **Upper Triangular**: A matrix is upper triangular if all entries below the main diagonal are zero. This matrix satisfies that condition.- **Lower Triangular**: A matrix is lower triangular if all entries above the main diagonal are zero. This matrix also satisfies that condition.- **Diagonal**: A diagonal matrix has all non-diagonal elements as zero, which is true for this matrix.- **Symmetric**: A matrix is symmetric if it equals its transpose. Since \( A = A^T \), it is symmetric.- **Skew Symmetric**: A matrix is skew symmetric if \( A = -A^T \), which is not true here as \( A = A^T \).

Conclusion on Matrix Properties

The matrix \[ A = \begin{bmatrix} 1 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & -1 \end{bmatrix} \]is diagonal, symmetric, upper triangular, and lower triangular. It is not skew symmetric.

Key concepts

Upper Triangular Matrix

An upper triangular matrix is a special type of square matrix where all the elements below its main diagonal are zero. To identify an upper triangular matrix, you just need to check if the values below the diagonal are zero. This characteristic makes performing calculations, like finding determinants, easier because those empty spaces eliminate unnecessary arithmetic. For instance, in a matrix:

• The first row can contain any numbers.
• The second row has zeros starting from the first element and may have numbers onwards.
• The third row starts with two zeros, with possible numbers onwards, and so on.

In our exercise example, the matrix \[A = \begin{bmatrix} 1 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & -1 \end{bmatrix}\]is upper triangular because all elements below its main diagonal are zero. Notice how easily you can perform operations on such matrices due to this structure.

Lower Triangular Matrix

A lower triangular matrix is another type of matrix that is "triangle-shaped" but in the opposite direction of an upper triangular matrix. Here, all entries above the main diagonal are zero. This type of matrix allows for similar computational conveniences, particularly in solving systems of linear equations or simplifying matrix factorizations.

• The first row only has the first element non-zero, while the rest are zeros.
• In the second row, the first two elements can have values, followed by zeros.
• In the third row and beyond, more of the initial elements may have non-zero values, with zeros following.

In the given exercise, the matrix \[A = \begin{bmatrix} 1 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & -1 \end{bmatrix}\]is also a lower triangular matrix. It fits the criteria as all entries above the diagonal are zero.

Diagonal Matrix

A diagonal matrix is a remarkable type of matrix where all elements outside the main diagonal are zero. Only the diagonal elements, from the top left to the bottom right, have non-zero values. This structure simplifies computations such as matrix multiplication because you only need to care about the main diagonal elements.
The matrix \[A = \begin{bmatrix} 1 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & -1 \end{bmatrix}\]in our exercise is clearly a diagonal matrix: all non-diagonal elements are zero. Another advantage of diagonal matrices is that they make it straightforward to compute powers of the matrix and solve certain sets of linear equations.

Symmetric Matrix

A symmetric matrix has a unique property: it is equal to its transpose. Transposing a matrix involves converting its rows into columns and vice versa. If upon transposing, the matrix remains unchanged, it is symmetric.
Such matrices have important applications, especially in physics and statistics, where symmetric properties lead to more predictable and often more theoretical solutions. An essential attribute is that all symmetric matrices must be square (same number of rows and columns).

• For example, swapping the first row and the first column results in no visible change.
• Similarly, the second row and column swap would be identical, and so forth.

Our example matrix,\[A = \begin{bmatrix} 1 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & -1 \end{bmatrix}\], is symmetric as it meets this property: \( A = A^T \). This symmetry simplifies understanding their structure and aids in computational tasks.