Question

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

Short answer

The transpose is the same zero matrix; it is symmetric, diagonal, upper and lower triangular.

Step-by-step solution

Identify Matrix Type

The given matrix is \(A = \begin{bmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{bmatrix}\), which is a 3x3 zero matrix. A zero matrix is a special type of matrix where every element is zero.

Transpose the Matrix

To find the transpose of matrix \(A\), denoted as \(A^T\), we swap its rows with columns. Since all elements of this matrix are zero, the transpose will remain the same: \[ A^T = \begin{bmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{bmatrix} \] Therefore, \(A^T = A\).

Analyze Matrix Properties

Examine the properties of the matrix: - **Lower Triangular:** All elements above the main diagonal are zero. True for this matrix. - **Upper Triangular:** All elements below the main diagonal are zero. True for this matrix. - **Diagonal:** All non-diagonal elements are zero. True for this matrix. - **Symmetric:** The matrix is equal to its transpose. True for this matrix. - **Skew Symmetric:** The transpose of the matrix is equal to the negative of the original matrix. False for this matrix since a zero matrix cannot be non-zero.

Key concepts

Zero Matrix

A zero matrix is a very special type of matrix where all elements are zeros. For instance, in our example, the matrix is \[A = \begin{bmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{bmatrix}.\] This means that no matter the size or dimensions of the zero matrix, every single entry is zero. This uniformity makes the zero matrix unique and versatile. A zero matrix is used as the additive identity in matrix addition, where adding any matrix to a zero matrix leaves the original matrix unchanged.
Some key points about zero matrices include:

• They serve as the identity for addition in matrix operations, similar to how 0 is the additive identity for real numbers.
• They are special cases of diagonal matrices, symmetric matrices, and triangular matrices.

Matrix Properties

Matrices have several important properties that help in understanding their structure and behavior. In the context of our example matrix, it possesses several notable properties:

• Diagonal Matrix: All elements outside the main diagonal are zero. Because the main diagonal also contains zeros, the zero matrix is technically considered a diagonal matrix.
• Triangular Matrix: There are two types of triangular matrices: upper and lower.
  • Upper Triangular: If all elements below the main diagonal are zero — a true statement for a zero matrix.
  • Lower Triangular: If all elements above the main diagonal are zero — also true here.
  Thus, the zero matrix is both upper and lower triangular.

These properties underline the flexibility of matrix formats and help us to quickly recognize and categorize matrices based on their structure.

Symmetric Matrix

A symmetric matrix is defined by its mirror-like property where the matrix is equal to its own transpose.
In mathematical terms, a matrix \(A\) is symmetric if: \[ A = A^T \] For our zero matrix, this symmetry is obvious, since transposing the matrix (flipping it over its diagonal) does not change any values—since all values are zero.
Some important characteristics of symmetric matrices include:

• They are always square matrices because this property only makes sense for a matrix with the same number of rows and columns.
• They are used in various mathematical and real-world applications, such as in solving systems of linear equations.
• A zero matrix, by being inherently unchanging when flipped, naturally fulfills the symmetric property.

Diagonal Matrix

A diagonal matrix is characterized by non-diagonal elements being zero. In our exercise, the zero matrix is a diagonal matrix by default. Every element, whether on or off the diagonal, is zero, but the defining factor is that it has zeros in off-diagonal positions.
Key features of diagonal matrices include:

• They are, by necessity, square matrices.
• They simplify operations like matrix multiplication. If a matrix is being multiplied by a diagonal matrix, only the diagonal entries from each matrix matter for the resulting product.
• The zero matrix is a special case where not only every off-diagonal is zero, every diagonal element is also zero, which makes it even simpler.

Diagonal matrices, including the zero matrix, are often used in linear algebra because of their simplicity and ease of computation. Their special structure allows for efficient algorithms in numerical computations.