# Section 6.3 Exercises

In each of the following, factor the matrix

*A*into a product $XD{X}^{-1}$, where*D*is diagonal:$A=\left[\begin{array}{rr}0& 1\\ 1& 0\end{array}\right]$

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

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

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

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

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

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

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

For each of the following, find a matrix

*B*such that ${B}^{2}=A$:$A=\left[\begin{array}{rr}2& 1\\ -2& -1\end{array}\right]$

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

Let

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

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

For each of the following, ...

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