Question

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{cc}-5 & -9 \\ 3 & 1 \\ -10 & -8\end{array}\right]\)

Short answer

\(A^T\) is \(\begin{bmatrix}-5 & 3 & -10\\ -9 & 1 & -8\end{bmatrix}\); \(A\) cannot have special properties like triangular or symmetric.

Step-by-step solution

Understanding the Transpose

The transpose of a matrix is obtained by swapping its rows with columns. For a matrix \(A\) with elements \(a_{ij}\), its transpose \(A^T\) will have elements \(a_{ji}\).

Identify the Dimensions of Matrix A

Matrix \(A\) is a 3x2 matrix, which means it has 3 rows and 2 columns. The transpose \(A^T\) of this matrix will have 2 rows and 3 columns, making it a 2x3 matrix.

Compute A^T

To find \(A^T\), swap the rows with columns: - First row of \(A\): \([-5, -9]\) becomes the first column of \(A^T\): \([-5, 3, -10]^T\).- Second row of \(A\): \([3, 1]\) becomes the second column of \(A^T\): \([-9, 1, -8]^T\).Thus, \(A^T = \begin{bmatrix}-5 & 3 & -10\ -9 & 1 & -8\end{bmatrix}\).

Analyze Matrix A

Matrix \(A\) is not square (3x2), and hence it cannot be triangular, diagonal, symmetric, or skew symmetric. These properties apply specifically to square matrices.

Key concepts

Triangular Matrix

A triangular matrix is a type of square matrix where all the elements above or below the main diagonal are zero. A square matrix can either be an upper triangular matrix, where all elements below the main diagonal are zero, or a lower triangular matrix, where all elements above the main diagonal are zero.
Upper and lower triangular matrices are useful in various mathematical computations, including solutions of linear equations and matrix decompositions.

• In an upper triangular matrix, if you have a 3x3 matrix, only the diagonal and above elements can be non-zero.
• In a lower triangular matrix, only the diagonal and below elements can be non-zero while above the diagonal remains zero.

Remember, being triangular is a property exclusive to square matrices, meaning this does not apply to the given 3x2 matrix in the problem, as it is not square.

Diagonal Matrix

A diagonal matrix is a special type of square matrix where all the elements off the main diagonal are zero. This means only the diagonal elements, such as those in a 3x3 matrix, can be non-zero.

• For example, in a 3x3 matrix, elements at positions (1,1), (2,2), and (3,3) can have non-zero values, while all other positions must be zero.

Diagonal matrices are crucial due to their simplicity and their role in simplifying matrix operations. They are commonly used in linear algebra to diagonalize matrices and simplify complex equations.
It's important to note that since the matrix in the exercise isn't square, it cannot be a diagonal matrix.

Symmetric Matrix

A symmetric matrix is a type of square matrix that is equal to its transpose. This means that the element at row i and column j is equal to the element at row j and column i, or mathematically, for a matrix A, \( A_{ij} = A_{ji} \). This property makes symmetric matrices particularly simple to work with in many scenarios like engineering and physics.

• An example of a symmetric matrix is: \( \begin{bmatrix} 2 & 3 & 4 \ 3 & 5 & 6 \ 4 & 6 & 8 \end{bmatrix} \)

Because symmetric matrices must be square, again, the 3x2 matrix in the exercise does not meet this requirement, so it cannot be symmetric.

Skew Symmetric Matrix

A skew symmetric matrix is a square matrix whose transpose is equal to its negative. This means for a matrix A, \( A^T = -A \). A key property of skew symmetric matrices is that all diagonal elements must be zero, making them quite unique compared to other matrix types.

• For instance, a skew symmetric matrix example is: \( \begin{bmatrix} 0 & -b & -c \ b & 0 & -f \ c & f & 0 \end{bmatrix} \)

Skew symmetric matrices arise in rotations and many applications in theoretical physics.
Like the other matrix types, skew symmetry is only possible with square matrices. Therefore, the non-square matrix in the original exercise cannot be skew symmetric.