91Ó°ÊÓ

TRUE OR FALSE

15. If matrix A is diagonalizable, then its transpose A^T must be diagonalizable as well.

Consider an n × n matrix A. A subspace V of ${\mathit{R}}^{\mathbf{n}}$ is said to be A invariant if $\mathit{A}\stackrel{\mathbf{→}}{\mathbf{v}}$is in V for all $\stackrel{\mathbf{→}}{\mathbf{v}}$in V. Describe all the one-dimensional A-invariant subspaces of Rⁿ , in terms of the eigenvectors of A.

TRUE OR FALSE

16. If A and B are two 3x3 matrices such that $\mathit{t}\mathit{r}\mathit{A}\mathbf{=}\mathit{t}\mathit{r}\mathit{B}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{d}\mathit{e}\mathit{t}\mathit{A}\mathbf{=}\mathit{d}\mathit{e}\mathit{t}\mathit{B}\mathbf{,}$then A and B must have the same eigenvalues.

TRUE OR FALSE

17. If 1 is the only eigenvalue of a $\mathit{n}\mathbf{×}\mathit{n}$ matrixA, then A must be l_n.

TRUE OR FALSE

18. If A and B are nxn matrices, if $\mathit{Î\pm }$is an eigenvalue of A, and if $\mathit{β}$is an eigenvalue of B, then $\mathit{Î\pm }\mathit{β}$ must be an eigenvalue of AB.

TRUE OR FALSE

19. If 3is an eigenvalue of an nxn matrix A, then 9 must be an eigenvalue of A².

Three holy men (let’s call them Anselm, Benjamin, and Caspar) put little stock in material things; their only earthly possession is a small purse with a bit of gold dust. Each day they get together for the following bizarre bonding ritual: Each of them takes his purse and gives his gold away to the two others, in equal parts. For example, if Anselm has 4 ounces one day, he will give 2 ounces each to Benjamin and Caspar.

(a) If Anselm starts out with 6 ounces, Benjamin with 1 ounce, and Caspar with 2 ounces, find formulas for the amounts a(t), b(t), and c(t) each will have after tdistributions.

Hint: The vector $\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]\mathbf{,}\left[\begin{array}{c}1\\ -1\\ 0\end{array}\right]\mathbf{}\mathbf{and}\mathbf{}\left[\begin{array}{c}1\\ 0\\ -1\end{array}\right]$, and will be useful.

(b) Who will have the most gold after one year, that is, after 365 distributions?

If 0 is an eigenvalue of a matrix A, then det A = 0.

TRUE OR FALSE

20. The matrix of any orthogonal projection onto a subspace V of Rⁿ is diagonalizable.

TRUE OR FALSE

21. All diagonalizable matrices are invertible.