Question

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{ccc}-3 & -4 & -5 \\ 0 & -3 & 5 \\ 0 & 0 & -3\end{array}\right]\)

Short answer

The transpose is \(A^{T} = \begin{bmatrix} -3 & 0 & 0 \\ -4 & -3 & 0 \\ -5 & 5 & -3 \end{bmatrix}\), and \(A\) is upper triangular.

Step-by-step solution

Identify the Matrix

The given matrix is \[ A = \begin{bmatrix} -3 & -4 & -5 \ 0 & -3 & 5 \ 0 & 0 & -3 \end{bmatrix} \]. This matrix is 3x3.

Determine the Transpose

The transpose of a matrix is found by swapping its rows with columns. The first row \([-3, -4, -5]\) becomes the first column, the second row \([0, -3, 5]\) becomes the second column, and the third row \([0, 0, -3]\) becomes the third column. Therefore, the transpose is: \[ A^{T} = \begin{bmatrix} -3 & 0 & 0 \ -4 & -3 & 0 \ -5 & 5 & -3 \end{bmatrix} \].

Analyze the Matrix Type

Compare the matrix \(A\) against the criteria for special matrices:1. **Upper Triangular:** Matrix \(A\) is upper triangular because all elements below the diagonal are zero.2. **Lower Triangular:** Not applicable because \(A\) has non-zero elements above the diagonal.3. **Diagonal:** Not applicable because it contains non-zero elements both above and below the diagonal that are not on the diagonal itself.4. **Symmetric:** A symmetric matrix is equal to its transpose. \(A eq A^{T}\) so it is not symmetric.5. **Skew Symmetric:** A skew-symmetric matrix satisfies \(A = -A^{T}\). This does not apply here.

Conclusion

The matrix \(A\) is an upper triangular matrix with its transpose \(A^{T}\) being \[ \begin{bmatrix} -3 & 0 & 0 \ -4 & -3 & 0 \ -5 & 5 & -3 \end{bmatrix} \].

Key concepts

Upper Triangular Matrix

An upper triangular matrix is a special type of square matrix where all the elements below the main diagonal are zero. This kind of matrix is useful in different areas of matrix algebra, such as solving linear systems and simplifying complex matrix operations. The main diagonal is the line of elements that goes from the top left to the bottom right of a square matrix.

When a matrix is in upper triangular form, operations like matrix multiplication and finding determinants become easier. For instance, the determinant of an upper triangular matrix can be computed by simply multiplying the diagonal elements together. This is a major benefit when working with large matrices.

In practical terms, consider the matrix \( A = \begin{bmatrix} -3 & -4 & -5 \ 0 & -3 & 5 \ 0 & 0 & -3 \end{bmatrix} \). You will notice that all the elements below the main diagonal are zero. This makes \( A \) a classic example of an upper triangular matrix, which simplifies many calculations you might perform on it.

Matrix Symmetry

Matrix symmetry is when a matrix is equal to its transpose, or \( A = A^{T} \). In other words, if you switch the rows and columns of a symmetric matrix, it remains unchanged. Symmetric matrices have various interesting properties that make them valuable in different mathematical applications.

For example, in symmetrical matrices, the diagonal elements are arbitrary, but the elements on either side are mirror images. This specific characteristic reduces the complexity of computations, since only half of the matrix (along with the diagonal) needs to be evaluated. Symmetric matrices also crop up frequently in fields like physics and statistics, owing to their predictive qualities.

Returning to our original exercise, the matrix \( A \) was not symmetric because \( A eq A^{T} \). While many famous matrices in algebra maintain symmetry, our matrix with its distinct rows and columns does not satisfy this relaxed condition.

Matrix Algebra Basics

Matrix algebra forms the foundation for many advanced mathematics topics, including linear transformations and systems of equations. The basic operations—addition, subtraction, multiplication, and finding the transpose—are crucial skills one needs to tackle more complex algebraic problems.

- **Addition and Subtraction:** Only matrices of the same dimensions can be added or subtracted. This involves performing the operation element-wise.- **Multiplication:** Matrix multiplication involves the dot product of rows and columns, resulting in complex calculations but offering powerful solutions, such as determining system behaviors.

- **Transpose:** Switching the rows and columns of a matrix gives us the matrix transpose, which is particularly useful for tasks like solving equations and transforming coordinate systems.

Our matrix \( A \) \( = \begin{bmatrix} -3 & -4 & -5 \ 0 & -3 & 5 \ 0 & 0 & -3 \end{bmatrix} \) exhibits some of these algebraic properties. Specifically, its transpose \( A^{T} \) \( = \begin{bmatrix} -3 & 0 & 0 \ -4 & -3 & 0 \ -5 & 5 & -3 \end{bmatrix} \) highlights an important aspect of matrix manipulation.

Diagonal Matrix

A diagonal matrix is another special kind of matrix where all the non-diagonal elements are zero. This means that the only elements that may be non-zero are those along the main diagonal from the top-left to the bottom-right.

Diagonal matrices are incredibly useful given their simplicity and the ease with which they can be manipulated. For instance, multiplying two diagonal matrices results in another diagonal matrix. Similarly, finding the inverse is straightforward if the diagonal elements are non-zero.

Although we analyzed the matrix \( A \) in the original exercise, we determined it was not a diagonal matrix because it had non-zero elements located above the diagonal line. This deviation from the zero-filled non-diagonal positions prevents \( A \) from fitting the definition of a diagonal matrix. Even so, recognizing when a matrix is diagonal is crucial, especially in fields that leverage large datasets and require efficient computations.