91Ó°ÊÓ

Prove that if \(A\) is an orthogonal matrix, then so are \(A^{T}\) and \(A^{-1}\).

For an invertible matrix \(A,\) prove that \(A\) and \(A^{-1}\) have the same eigenvectors. How are the eigenvalues of \(A\) related to the eigenvalues of \(A^{-1} ?\)

Prove that \(A\) and \(A^{T}\) have the same eigenvalues. Are the eigenspaces the same?

Consider the matrix below. $$A=\left[\begin{array}{rrrrr} -1 & 0 & -1 & 0 & 1 \\ 0 & 1 & 0 & -1 & 0 \\ -1 & 0 & 1 & 0 & -1 \\ 0 & -1 & 0 & -1 & 0 \\ 1 & 0 & -1 & 0 & -1 \end{array}\right]$$ (a) Is \(A\) symmetric? Explain. (b) Is \(A\) diagonalizable? Explain. (c) Are the eigenvalues of \(A\) real? Explain. (d) The eigenvalues of \(A\) are distinct. What are the dimensions of the corresponding eigenspaces? Explain. (e) Is \(A\) orthogonal? Explain. (f) For the eigenvalues of \(A\), are the corresponding eigenvectors orthogonal? Explain. (g) Is \(A\) orthogonally diagonalizable? Explain.

Find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject to the constraint. $$z=x_{1}^{2}+12 x_{2}^{2} ; 4 x_{1}^{2}+25 x_{2}^{2}=100$$

Prove that the constant term of the characteristic polynomial is \(\pm|A|\).

Prove that if \(A^{2}=O,\) then 0 is the only eigenvalue of \(A\) Getting Started: You need to show that if there exists a nonzero vector \(\mathbf{x}\) and a real number \(\lambda\) such that \(A \mathbf{x}=\lambda \mathbf{x},\) then if \(A^{2}=O, \lambda\) must be zero. (i) \(A^{2}=A \cdot A,\) so you can write \(A^{2} \mathbf{x}\) as \(A(A \mathbf{x})\) (ii) Use the fact that \(A \mathbf{x}=\lambda \mathbf{x}\) and the properties of matrix multiplication to show that \(A^{2} \mathbf{x}=\lambda^{2} \mathbf{x}\) (iii) \(A^{2}\) is a zero matrix, so you can conclude that \(\lambda\) must be zero.

Prove that the multiplicity of an eigenvalue is greater than or equal to the dimension of its eigenspace.

Find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject to the constraint. $$z=6 x_{1} x_{2} ;\|\mathbf{x}\|^{2}=1$$

(a) Explain how to model population growth using an age transition matrix and an age distribution vector, and how to find a stable age distribution vector. (b) Explain how to use a matrix equation to solve a system of first-order linear differential equations. (c) Explain how to use the Principal Axes Theorem to perform a rotation of axes for a conic and a quadric surface. (d) Explain how to solve a constrained optimization problem.