3.3 Inverses

1. Find the inverse of the following matrices, if they exist.

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

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

$B=\left[\begin{array}{lll}3& -1& 0\\ 2& 5& 1\end{array}\right]$

B has no inverse.

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

$C^{-1}=\left[\begin{array}{ccc}-1& 2& -2\\ 2& -1& 2\\ -2& 1& -1\end{array}\right]$

$D=\left[\begin{array}{cc}-3& 6\\ 1& -2\end{array}\right]$

D has no inverse.

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

$E^{-1}=\left[\begin{array}{ccc}2& 4& 2\\ -3& -2& -1\\ 1& 2& 1\end{array}\right]$

$F=\left[\begin{array}{cccc}1& 3& -2& 0\\ 0& 1& 1& 1\\ 0& 0& 1& -4\\ 0& 0& 0& 1\end{array}\right]$

$F^{-1}=\left[\begin{array}{cccc}1& -3& -5& -1\\ 0& 1& -1& -5\\ 0& 0& 1& 4\\ 0& 0& 0& 1\end{array}\right]$

2. Find the solution to the following linear system using inverses:

$x_1-2x_3=5$ $2x_1+x_2+2x_3=1$ $2x_1+x_2+3x_3=-2$

$x=\left[\begin{array}{c}1\\ -1\\ -2\end{array}\right]$

3. Find the inverse of the linear transformation…

$T:{\mathbf{R}}^3\to {\mathbf{R}}^3$ and use it to find a vector $\mathbf{x}$ such that $T(\mathbf{x})=\mathbf{y}$ where

$T\left(\left[\begin{array}{l}{x}_1\\ x_2\\ x_3\end{array}\right]\right)=\left[\begin{array}{c}{x}_1+3x_2+x_3\\ x_2+x_3\\ 2x_1-3x_3\end{array}\right]$

$T^{-1}\left(\left[\begin{array}{l}{y}_1\\ y_2\\ y_3\end{array}\right]\right)=\left[\begin{array}{c}\frac{3y_1+3y_2+y_3}{6}\\ y_2+y_3\\ \frac{y_1-3y_2-y_3}{2}\end{array}\right]$

a. $\mathbf{y}=\left[\begin{array}{c}0\\ -2\\ 2\end{array}\right]$

$\mathbf{x}=\left[\begin{array}{c}-1\\ -2\\ 2\end{array}\right]$

b. $\mathbf{y}=\left[\begin{array}{l}1\\ 3\\ 5\end{array}\right]$

$\mathbf{x}=\left[\begin{array}{c}-1/3\\ 8\\ -7/2\end{array}\right]$

4. Use a partitioning of $A$ to find the inverse.

$A=\left[\begin{array}{lll}2& 0& 0\\ 0& 5& 4\\ 0& 3& 2\end{array}\right]$

$A^{-1}=\left[\begin{array}{ccc}1/2& 0& 0\\ 0& 2/5& -4/5\\ 0& -3/5& 5/5\end{array}\right]$

5. Suppose that $A=PD{P}^{-1}$, where all matrices are square. Find an expression for each of $A^2$ and $A^3$, and then give a general formula for $A^n$.

$$A^2=P{D}^2{P}^{-1}$$ $$A^3=P{D}^3{P}^{-1}$$ $$A^n=P{D}^n{P}^{-1}$$