Use the matrix P to determine if the matrices A and A' are similar. P=begin{bmatrix}-1 & -1 1& 2 end{bmatrix}, A=begin{bmatrix}14 & 9 -20 & -13 end{bmatrix} text{ and } A'=begin{bmatrix}3 & -2 2 & -2 end{bmatrix} P^{-1}=? P^{-1}AP=? Are they similar? "Yes, they are similar" or "No, they are not similar"

Question
Matrices
asked 2020-10-26
Use the matrix P to determine if the matrices A and A' are similar.
\(P=\begin{bmatrix}-1 & -1 \\1& 2 \end{bmatrix}, A=\begin{bmatrix}14 & 9 \\-20 & -13 \end{bmatrix} \text{ and } A'=\begin{bmatrix}3 & -2 \\2 & -2 \end{bmatrix}\)
\(P^{-1}=?\)
\(P^{-1}AP=?\)
Are they similar?
"Yes, they are similar" or "No, they are not similar"

Answers (1)

2020-10-27
Step 1
Given:
\(P=\begin{bmatrix}-1 & -1 \\1& 2 \end{bmatrix}, A=\begin{bmatrix}14 & 9 \\-20 & -13 \end{bmatrix} \text{ and } A'=\begin{bmatrix}3 & -2 \\2 & -2 \end{bmatrix}\)
If \(P^{-1}AP= A'\) , then the matrices A and A' are similar.
Step 2
First, find \(P^{-1}\) as shown below.
\(P^{-1}=\frac{1}{det(P)} \cdot Adj(P)\)
\(=\frac{1}{\begin{vmatrix}-1 & -1 \\1 & 2 \end{vmatrix}} \cdot \begin{bmatrix}2 & 1 \\-1 & -1 \end{bmatrix}\)
\(=\frac{1}{-2+1} \cdot \begin{bmatrix}2 & 1 \\-1 & -1 \end{bmatrix}\)
\(=(-1) \cdot \begin{bmatrix}2 & 1 \\-1 & -1 \end{bmatrix}\)
\(=\begin{bmatrix}-2 & -1 \\1 & 1 \end{bmatrix}\)
Step 3
Compute \(P^{-1}AP\) as follows.
\(P^{-1}AP=\begin{bmatrix}-2 & -1 \\1 & 1 \end{bmatrix}\begin{bmatrix}14 & 9 \\-20 & -13 \end{bmatrix}\begin{bmatrix}-1 & -1 \\1 & 2 \end{bmatrix}\)
\(=\begin{bmatrix}-28+20 & -18+13 \\ 14-20 & 9-13 \end{bmatrix}\begin{bmatrix}-1 & -1 \\1 & 2 \end{bmatrix}\)
\(=\begin{bmatrix}-8 & -5 \\-6 & -4 \end{bmatrix}\begin{bmatrix}-1 & -1 \\1 & 2 \end{bmatrix}\)
\(=\begin{bmatrix}8-5 & 8-10 \\6-4 & 6-8 \end{bmatrix}\)
\(=\begin{bmatrix}3 & -2 \\2 & -2 \end{bmatrix}\)
Here, \(P^{-1}AP=A'\)
That implies, the matrices A and A' are similar.
Step 4
Therefore,
\(P^{-1}=\begin{bmatrix}-2 & -1 \\1 & 1 \end{bmatrix}\) and \(P^{-1}AP=\begin{bmatrix}3 & -2 \\2 & -2 \end{bmatrix}\)
And, the correct option is, "Yes, they are similar".
0

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