Chapter 7.5, Problem 1TFR

True-False Review

For Questions (a)-(h), decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem in fact from the text. If false, provide an example, illustration, or brief explanation of why the statement is false.

(a) If $A$ is an $n\times n$ orthogonal matrix, then $A$ is invertible and $A{A}^T=A^{T}A=I_n$.

(b) A real matrix $A$ whose characteristic polynomial is $p\left(\lambda \right)={\lambda }^3+\lambda$ cannot be symmetric.

(c) A real matrix $A$ with eigenvectors $v_1=\left[\begin{array}{c}1\\ 2\end{array}\right]$ and $v_3=\left[\begin{array}{c}1\\ 3\end{array}\right]$ cannot be symmetric.

(d) If $A$ is an $n\times n$ real, symmetric matrix with eigenvalues ${\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_n$ corresponding to a complete orthonormal set of eigenvectors for $A$, then the quadratic form $x^{T}Ax$ is transformed into ${\lambda }_1{y}_1^2+{\lambda }_2{y}_2^2+\cdots +{\lambda }_n{y}_n^2$.

(e) For any $n\times n$ real, symmetric matrix $A$ and vectors $v$ and $w$ in ${ℝ}^n$, we have $\langle Av,w\rangle =\langle v,Aw\rangle$.

(f) If $A$ and $B$ are $n\times n$ orthogonal matrices, then $AB$ is also an orthogonal matrix.

(g) A real $n\times n$ matrix $A$ is orthogonal if and only if its row (or column) vectors are orthogonal unit vectors.

(h) Any real matrix with a complete set of orthonormal eigenvectors is symmetric.