Lipossig

2020-10-26

Use the matrix P to determine if the matrices A and A

estenutC

Step 1
Given:

If ${P}^{-1}AP={A}^{\prime }$ , then the matrices A and A' are similar.
Step 2
First, find ${P}^{-1}$ as shown below.
${P}^{-1}=\frac{1}{det\left(P\right)}\cdot Adj\left(P\right)$
$=\frac{1}{|\begin{array}{cc}-1& -1\\ 1& 2\end{array}|}\cdot \left[\begin{array}{cc}2& 1\\ -1& -1\end{array}\right]$
$=\frac{1}{-2+1}\cdot \left[\begin{array}{cc}2& 1\\ -1& -1\end{array}\right]$
$=\left(-1\right)\cdot \left[\begin{array}{cc}2& 1\\ -1& -1\end{array}\right]$
$=\left[\begin{array}{cc}-2& -1\\ 1& 1\end{array}\right]$
Step 3
Compute ${P}^{-1}AP$ as follows.
${P}^{-1}AP=\left[\begin{array}{cc}-2& -1\\ 1& 1\end{array}\right]\left[\begin{array}{cc}14& 9\\ -20& -13\end{array}\right]\left[\begin{array}{cc}-1& -1\\ 1& 2\end{array}\right]$
$=\left[\begin{array}{cc}-28+20& -18+13\\ 14-20& 9-13\end{array}\right]\left[\begin{array}{cc}-1& -1\\ 1& 2\end{array}\right]$
$=\left[\begin{array}{cc}-8& -5\\ -6& -4\end{array}\right]\left[\begin{array}{cc}-1& -1\\ 1& 2\end{array}\right]$
$=\left[\begin{array}{cc}8-5& 8-10\\ 6-4& 6-8\end{array}\right]$
$=\left[\begin{array}{cc}3& -2\\ 2& -2\end{array}\right]$
Here, ${P}^{-1}AP={A}^{\prime }$
That implies, the matrices A and A' are similar.
Step 4
Therefore,
${P}^{-1}=\left[\begin{array}{cc}-2& -1\\ 1& 1\end{array}\right]$ and ${P}^{-1}AP=\left[\begin{array}{cc}3& -2\\ 2& -2\end{array}\right]$
And, the correct option is, "Yes, they are similar".

Jeffrey Jordon