Question

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{ccc}-9 & 4 & 10 \\ 6 & -3 & -7 \\ -8 & 1 & -1\end{array}\right]\)

Short answer

The transpose of matrix A is \(A^T = \begin{bmatrix}-9 & 6 & -8 \\ 4 & -3 & 1 \\ 10 & -7 & -1\end{bmatrix}\), and it is neither triangular, diagonal, symmetric, nor skew symmetric.

Step-by-step solution

Definition of Transpose

To find the transpose of a matrix, we swap its rows with its columns. This means the element at position (i, j) in the original matrix will move to position (j, i) in the transposed matrix.

Apply Transpose to Matrix A

Given the matrix \(A\): \[A = \begin{bmatrix}-9 & 4 & 10 \6 & -3 & -7 \-8 & 1 & -1\end{bmatrix}\]Transpose \(A\) to get \(A^T\): \[A^T = \begin{bmatrix}-9 & 6 & -8 \4 & -3 & 1 \10 & -7 & -1\end{bmatrix}\]

Check for Upper/Lower Triangular

A matrix is upper triangular if all the elements below the main diagonal are zero. It is lower triangular if all the elements above the main diagonal are zero. In this case, \(A\) has non-zero elements both above and below the main diagonal.

Check for Diagonal Matrix

A diagonal matrix has non-zero elements only on its main diagonal, with all other elements being zero. Matrix \(A\) has non-diagonal elements that are not zero, so it is not a diagonal matrix.

Check for Symmetry

A symmetric matrix satisfies \(A = A^T\). Compare \(A\) and \(A^T\) and see if they are equal. In this case, they are not equal, so \(A\) is not symmetric.

Check for Skew Symmetry

A skew-symmetric matrix satisfies \(A = -A^T\). Compare \(A\) with \(-A^T\) and see if they are equal. They are not, so \(A\) is not skew symmetric.

Key concepts

Upper Triangular Matrix

An upper triangular matrix is characterized by having all zero entries below the main diagonal. The main diagonal of a matrix is the set of elements that stretch from the top-left to the bottom-right of the square matrix.
In an upper triangular matrix, any element in positions where the row number is greater than the column number must be zero.

• For example, in a 3x3 matrix, the elements in positions (2,1) and (3,1), and (3,2) should all be zero for it to be considered upper triangular.
• This structure simplifies many matrix operations, such as solving systems of equations, because you can work starting from the top without backward substitution.

Determining if a matrix is upper triangular is straightforward: simply check the entries below the main diagonal. If any of them are non-zero, the matrix is not upper triangular.

Symmetric Matrix

A symmetric matrix is one that is equal to its transpose. This means the element at position (i, j) in the original matrix is the same as the element at position (j, i) in the transposed matrix.
Because of this property:

• A symmetric matrix must always be square.
• The main diagonal of a symmetric matrix must have real numbers since they must remain unchanged during transposition.
• Example: If element (1,2) is 5, then element (2,1) must also be 5.

Symmetric matrices have applications in various fields, including physics and engineering, where they are often used to represent systems that are inherently balanced or have a specific type of natural equilibrium.

Diagonal Matrix

A diagonal matrix is unique because all of its non-diagonal elements are zero. Only the entries on the main diagonal can be non-zero.
This kind of matrix is particularly useful because:

• Diagonal matrices are easy to deal with mathematically; operations like taking powers or finding determinants become straightforward.
• They are used in various transformations and can simplify complex operations.

In any square diagonal matrix, the only elements that can differ from zero must lie on the main diagonal. Hence, it's quick to verify whether a matrix is diagonal – just check if all non-diagonal positions are zero! This efficiency explains why diagonal matrices play such an essential role in computational mathematics.

Skew-Symmetric Matrix

A skew-symmetric matrix has a special property that makes it quite unique: the transpose of the matrix is equal to its negative. In formal terms, for a matrix to be skew-symmetric, it should satisfy the condition:

• \( A = -A^T \)
• This implies that all elements on the main diagonal must be zero because they would need to be their own negatives, which is only true for zero.
• It also implies that an element and its transposed counterpart must sum to zero, i.e., \( a_{ij} = -a_{ji} \) for all i ≠ j.

Many properties of skew-symmetric matrices make them useful in physics, particularly in understanding rotational motion and complex networks.