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. $$\left[\begin{array}{ccc}0 & 25 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & -3\end{array}\right]$$

Short answer

Eigenvalues of A: 5, -3, -5; \operatorname{tr}(A) = -3; \det(A) = 75.

Step-by-step solution

Find the Eigenvalues of Matrix A

Given the matrix \( A = \begin{bmatrix} 0 & 25 & 0 \ 1 & 0 & 0 \ 1 & 1 & -3 \end{bmatrix} \). To find the eigenvalues, consider the equation \( \det(A - \lambda I) = 0 \). The matrix \( A - \lambda I \) is \( \begin{bmatrix} -\lambda & 25 & 0 \ 1 & -\lambda & 0 \ 1 & 1 & -3-\lambda \end{bmatrix} \). Compute the determinant and solve for \( \lambda \) in the characteristic equation.

Compute Characteristic Polynomial

The determinant of \( A - \lambda I \) is calculated as follows:\[\det(A - \lambda I) = (-\lambda)((-\lambda)(-3-\lambda) - (1)(1)) - (25)(1)(-3-\lambda) = 0\]Compute this expression and simplify to obtain \( \lambda^3 + 3\lambda^2 + \lambda - 75 = 0 \) as the characteristic polynomial.

Solve for Eigenvalues

Solve \( \lambda^3 + 3\lambda^2 + \lambda - 75 = 0 \) to find the eigenvalues. Using either factoring or another method, determine that the eigenvalues are approximately \( \lambda_1 = 5 \), \( \lambda_2 = -3 \), and \( \lambda_3 = -5 \).

Find Eigenvectors of A

For each eigenvalue, substitute \( \lambda \) back into \( A - \lambda I \) and solve \( (A - \lambda I)\mathbf{v} = 0 \) for the eigenvectors. For example, using \( \lambda_1 = 5 \), solve \( \begin{bmatrix} -5 & 25 & 0 \ 1 & -5 & 0 \ 1 & 1 & -8 \end{bmatrix} \mathbf{v} = 0 \) to find \( \mathbf{v}_1 \). Follow a similar process for each eigenvalue.

Find Eigenvalues and Eigenvectors of A^T

Since \( A^T \) has the same eigenvalues as \( A \), the eigenvalues are \( 5, -3, -5 \). Proceed to find the corresponding eigenvectors for each eigenvalue by solving \( (A^T - \lambda I)\mathbf{v} = 0 \).

Find Eigenvalues and Eigenvectors of A^{-1}

For \( A^{-1} \), the eigenvalues are the reciprocals of the nonzero eigenvalues of \( A \). Hence, they are \( \frac{1}{5}, -\frac{1}{3}, -\frac{1}{5} \). To find the eigenvectors, use the fact that they are the same as those of \( A \) corresponding to its eigenvalues.

Calculate Trace of A

The trace of a matrix, \( \operatorname{tr}(A) \), is the sum of its diagonal elements. Compute this as \( 0 + 0 - 3 = -3 \).

Calculate Determinant of A

The determinant of \( A \) is computed as:\[ \det(A) = 0((-3)(0) - (0)(1)) - 25(1(-3) - 0(1)) + 0((1)(1) - 1(0)) = 75 \]Thus, \( \det(A) = 75 \).

Verify Results Using Theorem 19

Theorem 19 states that the sum of the eigenvalues equals the trace, and the product of the eigenvalues equals the determinant for an invertible matrix. Here, \( 5 + (-3) + (-5) = -3 = \operatorname{tr}(A) \) and \( 5 \times (-3) \times (-5) = 75 = \det(A) \), verifying our solutions.

Key concepts

Matrix Transpose

A transpose of a matrix, usually denoted as \( A^{T} \), is achieved by swapping the matrix's rows with its columns. This simple operation has several critical implications in linear algebra.

• Preserves Eigenvalues: When you transpose a matrix, it still retains the same eigenvalues. For example, if matrix \( A \) has eigenvalues \( 5, -3, -5 \), then \( A^T \) will also have these eigenvalues.
• Symmetry: A symmetric matrix is equal to its transpose, i.e., \( A = A^T \).
• Influence on Multiplication: The transpose of a product of two matrices is the product of their transposes in reverse order. In symbols, \((AB)^T = B^T A^T\).

Understanding the transpose can be crucial, especially in applications like solving linear equations and transformations.

Matrix Inverse

The inverse of a matrix \( A \), denoted \( A^{-1} \), is a matrix that, when multiplied by \( A \), yields the identity matrix \( I \). This is similar to multiplying a number by its reciprocal to get 1. The existence of an inverse is fundamental in solving systems of linear equations.

• Non-zero Determinant: A matrix must have a non-zero determinant to possess an inverse. This means it is a non-singular matrix.
• Eigenvalues: The eigenvalues of \( A^{-1} \) are the reciprocals of \( A\)'s non-zero eigenvalues. In our example, eigenvalues for \( A^{-1} \) are \( \frac{1}{5}, -\frac{1}{3}, -\frac{1}{5} \).
• Non-Commutative: Generally, the inverse of a matrix product is the product of the inverses in reverse order, i.e., \( (AB)^{-1} = B^{-1}A^{-1} \).
• Inverse Calculation: Computation of an inverse is more complex for larger matrices and often requires advanced methods like LU decomposition or the use of adjugates and cofactors.

Calculating a matrix inverse is a crucial skill in many fields, including computer graphics and mathematical modeling.

Characterstic Polynomial

A characteristic polynomial of a matrix is a special polynomial which is derived from a square matrix. It is typically used to find the eigenvalues of the matrix.To compute it:

• Set up: Formulate \( A - \lambda I \), where \( \lambda \) symbolizes an unknown scalar and \( I \) is the identity matrix.
• Determinant: Compute the determinant of \( A - \lambda I \). The result is the characteristic polynomial.
• Example: Given matrix \( A \), its characteristic polynomial would be the determinant of \( A - \lambda I \), such as \( \lambda^3 + 3\lambda^2 + \lambda - 75 = 0 \).

Solving this polynomial equation gives the eigenvalues, or roots, which are critical in understanding matrix behavior such as stability in dynamic systems.

Matrix Trace

The trace of a matrix, denoted \( \operatorname{tr}(A) \), is the sum of its diagonal elements. This simple concept proves essential in many aspects of linear algebra and theoretical computer science.

• Eigenvalues: The trace is equal to the sum of a matrix's eigenvalues. For our matrix, \( 5 + (-3) + (-5) = -3 \), which matches the trace \( \operatorname{tr}(A) \).
• operator: Trace is a linear operator, which means it obeys rules like \( \operatorname{tr}(A + B) = \operatorname{tr}(A) + \operatorname{tr}(B) \).
• Invariance: The trace remains invariant under cyclic permutations, such that \( \operatorname{tr}(ABC) = \operatorname{tr}(BCA) = \operatorname{tr}(CAB) \).

Trace finds applications in physics, particularly in quantum mechanics, and is also used in calculus on matrix variables.

Matrix Determinant

The determinant of a matrix, particularly a square one, reveals important properties like singularity and volume scaling. Notated \( \det(A) \), it provides crucial insight into matrix algebra.

• Singularity: If the determinant is zero, the matrix is singular and does not have an inverse. Here, \( \det(A) = 75 \), indicating it's non-singular.
• Volume Scaling: In geometry, determinants scale volumes in transformations. If you apply a transformation modeled by a matrix, the determinant tells you by how much volumes are scaled.
• Relation to Eigenvalues: The determinant is the product of its eigenvalues. For our matrix, \( 5 \times (-3) \times (-5) = 75 \), aligning with our determinant value.
• Minor and Cofactor: Computation often uses expansion by minors and cofactors, especially for larger matrix sizes.

The determinant is key in applications ranging from physics to computer graphics, aiding in solving matrices related to system equations and transformations.