Question

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{ll}1 & 0 \\ 0 & 9\end{array}\right]\)

Short answer

The matrix is diagonal and symmetric. Its transpose is the same as the original matrix.

Step-by-step solution

Understand the Matrix

The given matrix is \[ A = \begin{bmatrix} 1 & 0 \ 0 & 9 \end{bmatrix} \]This matrix is square with dimensions 2x2.

Determine if the Matrix is Triangular or Diagonal

A matrix is upper triangular if all the entries below the main diagonal are zero. A matrix is lower triangular if all the entries above the main diagonal are zero. A matrix is diagonal if all the off-diagonal elements are zero. For matrix \(A\): - Entries below the diagonal are zero. - Entries above the diagonal are zero.Thus, matrix \(A\) is a diagonal matrix.

Check Symmetry Properties

A symmetric matrix is one where \(A = A^T\). A skew-symmetric matrix is one where \(A = -A^T\). Matrix \(A\) is:- Symmetric, because it equals its transpose (check this in the next step).- Not skew-symmetric, because all elements are not negated in the transpose.

Calculate the Transpose

To find the transpose \(A^T\), swap the rows and columns of \(A\). For matrix \(A\):\[ A = \begin{bmatrix} 1 & 0 \ 0 & 9 \end{bmatrix} \]The transpose is: \[ A^T = \begin{bmatrix} 1 & 0 \ 0 & 9 \end{bmatrix} \]The matrix remains unchanged, confirming that it is symmetric.

Key concepts

Symmetric Matrix

A symmetric matrix is a special type of square matrix which is mirrored along its main diagonal. This means the entries on the either side of the diagonal are equal. Mathematically, a matrix is symmetric if it satisfies the condition: \(A = A^T\).This means that when you transpose the matrix, you get the same matrix back.The main diagonal is the line of entries from the top left to the bottom right. In a symmetric matrix, the entry at position \((i, j)\) is the same as the entry at position \((j, i)\).For example:- If your matrix looks like \[ \begin{bmatrix} a & b \ b & c \end{bmatrix} \] then you know it is symmetric because \(a = a\), \(b = b\), and \(c = c\).Symmetric matrices are important in various fields such as physics, computer science, and statistics because they simplify computations and have desirable mathematical properties.

Diagonal Matrix

A diagonal matrix is a matrix in which all the elements outside the main diagonal are zero. This characteristic helps to quickly identify a diagonal matrix without performing any complex operations.The main diagonal extends from the top-left of the matrix to the bottom-right.In a diagonal matrix:

• The elements \((i, j) = 0\) whenever \(i eq j\).

To represent a diagonal matrix:\[ \begin{bmatrix} d_1 & 0 \0 & d_2 \0 & \cdots & 0 \0 & \ & \ & d_n \end{bmatrix} \]Diagonal matrices are considered simple among matrix types because their non-diagonal elements are zero. Their simplicity leads to an easily computable determinant and eigenvalues. They are highly beneficial in simplifying matrix operations, especially matrix multiplication and finding the inverse, where applicable.

Triangular Matrix

Triangular matrices can be either upper or lower triangular, based on where the non-zero elements are located concerning the main diagonal. Let's break these down:- **Upper Triangular Matrix**: This is when all the elements below the main diagonal are zero. Essentially, the non-zero entries fill only the upper half of the matrix.- **Lower Triangular Matrix**: Oppositely, this matrix has all elements above the main diagonal set to zero, having non-zero entries populate the lower half.An example of an upper triangular matrix might be:\[ \begin{bmatrix} x & y \ 0 & z \end{bmatrix} \]While a lower triangular matrix may look like:\[ \begin{bmatrix} p & 0 \ q & r \end{bmatrix} \]Triangular matrices are key in computational mathematics, including solving linear equations and eigenvalue computations. They make matrix operations more efficient since certain calculations can be skipped or simplified due to the presence of zero elements.

Matrix Properties

Matrices have numerous intriguing properties making them crucial tools in linear algebra and various real-world applications. Understanding their properties is essential for simplifying complex problems.Basic matrix properties involve operations such as addition, multiplication, and transposition. Here are some noteworthy properties:

• **Addition**: Two matrices can be added only if they have the same dimensions.
• **Multiplication**: Matrix multiplication is associative but not commutative. This means \((A \times B) eq (B \times A)\).
• **Identity Matrix**: Any square matrix multiplied by the identity matrix results in itself.
• **Transposition**: Switching rows with columns in a matrix forms its transpose.

Understanding these properties aids in comprehending more complex matrix problems. Matrices allow us to model and solve systems of linear equations, perform transformations in space, and enable powerful computational applications in science and engineering.