Linear Algebra with Applications, 10th Edition

Section 6.3 Exercises

1. In each of the following, factor the matrix A into a product $XD{X}^{-1}$, where D is diagonal:

   1. $A=\left[\begin{array}{rr}0& 1\\ 1& 0\end{array}\right]$

   2. $A=\left[\begin{array}{rr}5& 6\\ -2& -2\end{array}\right]$

   3. $A=\left[\begin{array}{rr}2& -8\\ 1& -4\end{array}\right]$

   4. $A=\left[\begin{array}{rrr}2& 2& 1\\ 0& 1& 2\\ 0& 0& -1\end{array}\right]$

   5. $A=\left[\begin{array}{rrr}1& 0& 0\\ -2& 1& 3\\ 1& 1& -1\end{array}\right]$

   6. $A=\left[\begin{array}{rrr}1& 2& -1\\ 2& 4& -2\\ 3& 6& -3\end{array}\right]$

2. For each of the matrices in Exercise 1, use the $XD{X}^{-1}$ factorization to compute $A^6$.

3. For each of the nonsingular matrices in Exercise 1, use the $XD{X}^{-1}$ factorization to compute $A^{-1}$.

4. For each of the following, find a matrix B such that $B^2=A$:

   1. $A=\left[\begin{array}{rr}2& 1\\ -2& -1\end{array}\right]$

   2. $A=\left[\begin{array}{rrr}9& -5& 3\\ 0& 4& 3\\ 0& 0& 1\end{array}\right]$

5. Let A be a nondefective $n\times n$ matrix with diagonalizing matrix X. Show that the matrix $Y=(X^{-1}{)}^T$ diagonalizes $A^T$.

6. Let A be a diagonalizable matrix whose eigenvalues are all either 1 or −1. Show that $A^{-1}=A$.

7. Show that any $3\times 3$ matrix of the form

   $$\left[\begin{array}{ccc}a& 1& 0\\ 0& a& 1\\ 0& 0& b\end{array}\right]$$

   is defective.

8. For each of the following, ...