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Magazine Chapter 4: Problem 4
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
Expert verified
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
01
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 \(\lambda\):\[\lambda_1 = 4, \quad \lambda_2 = 5 \]
02
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 \(\lambda_2 = 5\).
03
Finding Eigenvalues of A^T
The eigenvalues of \(A^T\) are the same as those of \(A\). Thus, they are \(\lambda_1 = 4\) and \(\lambda_2 = 5\).
04
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 \(\begin{bmatrix}8 \ 1\end{bmatrix}\).
05
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 \(\lambda_2 = \frac{1}{5}\).
06
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 \(\begin{bmatrix}8 \ 1\end{bmatrix}\).
07
Finding the Trace of A
The trace of a matrix is the sum of its diagonal elements. For matrix \(A\):\[\operatorname{tr}(A) = -4 + 13 = 9\]
08
Finding the Determinant of A
The determinant of the matrix \(A = \begin{bmatrix}-4 & 72 \ -1 & 13\end{bmatrix}\) is given by:\[\det(A) = (-4)(13) - (72)(-1) = -52 + 72 = 20\]
09
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
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 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 \( A \).
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:
\[ \operatorname{tr}(A) = a_{11} + a_{22} + \cdots + a_{nn} \]
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.
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:
\[ A = \begin{bmatrix} -4 & 72 \ -1 & 13 \end{bmatrix} \]
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.
\[ \det(A) = ad - bc \]
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.
\[ A = \begin{bmatrix} -4 & 72 \ -1 & 13 \end{bmatrix} \]
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.
