Question

A matrix \(A\) is given. For each, (a) Find the eigenvalues of \(A,\) and for each eigenvalue, find an eigenvector. (b) Do the same for \(A^{T}\). (c) Do the same for \(A^{-1}\). (d) Find \(\operatorname{tr}(A)\). (e) Find det \((A)\). Use Theorem 19 to verify your results. $$\left[\begin{array}{ll}-4 & 72 \\ -1 & 13\end{array}\right]$$

Short answer

Eigenvalues of \(A\) are 4 and 5. Eigenvectors for \(\lambda = 4\) and \(5\) are \([9, 1]^T\) and \([8, 1]^T\), respectively. \(\operatorname{tr}(A) = 9\) and \(\det(A) = 20\).

Step-by-step solution

Finding Eigenvalues of A

To find the eigenvalues of matrix \(A\), solve the characteristic equation \(\det(A - \lambda I) = 0\), where \(I\) is the identity matrix. For \(A = \begin{bmatrix}-4 & 72 \ -1 & 13\end{bmatrix}\), we set up the equation as follows:\[\det\begin{pmatrix}-4-\lambda & 72 \ -1 & 13-\lambda\end{pmatrix} = 0\]Compute the determinant:\[(-4-\lambda)(13-\lambda) - (-1)(72) = \lambda^2 - 9\lambda + 20 = 0\]Solve for \(\lambda\):\[\lambda_1 = 4, \quad \lambda_2 = 5 \]

Finding Eigenvectors of A

For each eigenvalue, substitute back into \((A - \lambda I)v = 0\) to find the eigenvectors.For \(\lambda_1 = 4\):\[\begin{bmatrix}-8 & 72 \ -1 & 9\end{bmatrix} \begin{bmatrix}x \ y\end{bmatrix} = \begin{bmatrix}0 \ 0\end{bmatrix}\]This simplifies to:\[x - 9y = 0\, \Rightarrow y = \frac{x}{9}\]Thus, the eigenvector is \(\begin{bmatrix}9 \ 1\end{bmatrix}\) for \(\lambda_1 = 4\).For \(\lambda_2 = 5\):\[\begin{bmatrix}-9 & 72 \ -1 & 8\end{bmatrix} \begin{bmatrix}x \ y\end{bmatrix} = \begin{bmatrix}0 \ 0\end{bmatrix}\]This simplifies to:\[x - 8y = 0\, \Rightarrow y = \frac{x}{8}\]Thus, the eigenvector is \(\begin{bmatrix}8 \ 1\end{bmatrix}\) for \(\lambda_2 = 5\).

Finding Eigenvalues of A^T

The eigenvalues of \(A^T\) are the same as those of \(A\). Thus, they are \(\lambda_1 = 4\) and \(\lambda_2 = 5\).

Finding Eigenvectors of A^T

For \(A^T = \begin{bmatrix}-4 & -1 \ 72 & 13\end{bmatrix}\), the eigenvectors are also the same as for \(A\) because the results are derived from the same characteristic polynomial.For \(\lambda_1 = 4\), the eigenvector is \(\begin{bmatrix}9 \ 1\end{bmatrix}\).For \(\lambda_2 = 5\), the eigenvector is \(\begin{bmatrix}8 \ 1\end{bmatrix}\).

Finding Eigenvalues of A^{-1}

The eigenvalues of \(A^{-1}\) are the reciprocals of the eigenvalues of \(A\). Therefore, they are \(\lambda_1 = \frac{1}{4}\) and \(\lambda_2 = \frac{1}{5}\).

Finding Eigenvectors of A^{-1}

The eigenvectors of \(A^{-1}\) are the same as the eigenvectors of \(A\) and \(A^T\). Thus, for \(\lambda_1 = \frac{1}{4}\), the eigenvector is \(\begin{bmatrix}9 \ 1\end{bmatrix}\), and for \(\lambda_2 = \frac{1}{5}\), the eigenvector is \(\begin{bmatrix}8 \ 1\end{bmatrix}\).

Finding the Trace of A

The trace of a matrix is the sum of its diagonal elements. For matrix \(A\):\[\operatorname{tr}(A) = -4 + 13 = 9\]

Finding the Determinant of A

The determinant of the matrix \(A = \begin{bmatrix}-4 & 72 \ -1 & 13\end{bmatrix}\) is given by:\[\det(A) = (-4)(13) - (72)(-1) = -52 + 72 = 20\]

Verification using Theorem 19

According to Theorem 19, for a square matrix \(A\), the eigenvalues \(\lambda\) satisfy \(\operatorname{tr}(A) = \sum \lambda_i\) and \(\det(A) = \prod \lambda_i\).For matrix \(A\):- Sum of eigenvalues: \(4 + 5 = 9\)- Product of eigenvalues: \(4 \times 5 = 20\)Both identities are verified, confirming the results.

Key concepts

Matrix Transpose

In mathematics, the transpose of a matrix is achieved by flipping the matrix over its diagonal. This operation swaps the row and column indices of each element of the matrix. If you have a matrix \( A = [a_{ij}] \), the transpose of \( A \), denoted as \( A^T \), is formed by transposing the i-th row and j-th column into the j-th row and i-th column.

For instance, consider the original matrix in this problem:

\[ A = \begin{bmatrix} -4 & 72 \ -1 & 13 \end{bmatrix} \]

The transpose, \( A^T \), becomes:

\[ A^T = \begin{bmatrix} -4 & -1 \ 72 & 13 \end{bmatrix} \]

This means that each element \( a_{ij} \) of the original matrix becomes \( a_{ji} \) in the transposed matrix.

• The eigenvalues of a matrix \( A \) and its transpose \( A^T \) are the same because they share the same characteristic polynomial.
• However, eigenvectors might differ unless the matrix is symmetric.

The matrix transpose is an essential tool in various mathematical computations, including solving systems of linear equations, in optimization, and in finding the determinant and eigenvalues.

Matrix Inverse

The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. A matrix \( A \) has an inverse, denoted by \( A^{-1} \), only if \( A \) is square (i.e., it has the same number of rows as columns) and its determinant is non-zero.

The main property of a matrix and its inverse is:

\[ A \times A^{-1} = A^{-1} \times A = I \]

where \( I \) is the identity matrix of the same size as \( A \). For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the inverse can be calculated using:

\[ A^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \ -c & a \end{bmatrix} \]

This formula applies only if the determinant \( ad-bc eq 0 \).

• The eigenvalues of \( A^{-1} \) are the reciprocals of the eigenvalues of \( A \).
• The eigenvectors remain the same as those of \( A \).

The concept of matrix inverse is fundamental in solving linear equations, particularly in finding solutions to systems where matrices represent transformations or coefficients.

Trace of a Matrix

The trace of a matrix is a very simple concept but quite useful in linear algebra. It is defined as the sum of the elements on the main diagonal of a square matrix. Symbolically, if \( A \) is a matrix, then the trace, denoted by \( \operatorname{tr}(A) \), is:

\[ \operatorname{tr}(A) = a_{11} + a_{22} + \cdots + a_{nn} \]

where \( a_{ii} \) represents the diagonal elements of matrix \( A \). For the matrix in our example:

\[ A = \begin{bmatrix} -4 & 72 \ -1 & 13 \end{bmatrix} \]

The trace is:

\[ \operatorname{tr}(A) = -4 + 13 = 9 \]

The trace of a matrix has some interesting properties and applications:

• It equals the sum of the eigenvalues of the matrix.
• It is invariant under matrix transpose (i.e., \( \operatorname{tr}(A) = \operatorname{tr}(A^T) \)).
• It provides insight into the matrix's eigenvalues and conjugate transpose relationships.

The trace helps to quickly verify calculations involving eigenvalues and can be part of more extensive theorems and properties used in matrix decompositions and other complex linear algebra applications.

Determinant of a Matrix

The determinant is a special figure calculated from a square matrix, which provides vital information about the matrix's properties. For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant, denoted \( \det(A) \), is calculated as:

\[ \det(A) = ad - bc \]

This value tells us several important things about the matrix:

• If the determinant is zero, the matrix is singular, meaning it doesn't have an inverse.
• A non-zero determinant indicates that the matrix is invertible.
• The determinant is also involved in the calculation for the area transformations in 2-dimensional space induced by the matrix.

For our specific example matrix:

\[ A = \begin{bmatrix} -4 & 72 \ -1 & 13 \end{bmatrix} \]

The determinant is:

\[ \det(A) = (-4)(13) - (72)(-1) = -52 + 72 = 20 \]

Determinants are extensively used in linear transformations, analysis of linear systems, and the theory of eigenvalues and eigenvectors, making them one of the core components of linear algebra.