How to determine two matrices are conjugate.

Which of the following statements are true?

• The matrices $A=\left[ {\begin{array}{cc} 1 & 1 \\ 0 & 1\\ \end{array} } \right]$ and $B=\left[ {\begin{array}{cc} 1 & 0 \\ 1 & 1\\ \end{array} } \right]$ are conjugate in $GL_2(\mathbb{R})$
• The matrices $A=\left[ {\begin{array}{cc} 1 & 1 \\ 0 & 1\\ \end{array} } \right]$ and $B=\left[ {\begin{array}{cc} 1 & 0 \\ 1 & 1\\ \end{array} } \right]$ are conjugate in $SL_2(\mathbb{R})$
• The matrices $C=\left[ {\begin{array}{cc} 1 & 0 \\ 0 & 2\\ \end{array} } \right]$ and $D=\left[ {\begin{array}{cc} 1 & 3 \\ 0 & 2\\ \end{array} } \right]$ are conjugate in $GL_2(\mathbb{R})$

I know the conjugate matrices have the same eigenvalues. But all of this matrices have the same eigenvalues. I am not sure how to determine that which matrices are conjugate to each other.

The definition of conjugacy is: if $A$ and $B$ are two matrices such that $$A= P^{-1}BP$$ then they are conjugate. In $GL(2,\mathbb{R})$ is the same as similarity so not only the eigenvalues have to be preserved: mainly two similar matrices in $GL(2,\mathbb{R})$ have the same Jordan normal form
Notice that A is in Jordan canonical form (just look at the eigenvalues). Notice that the $D$ matrix is diagonalisable
Probabily this forum Why aren't this two matrices conjugate in $SL(2,\mathbb{R})$ can help for the second question.
• What is $D$ here? – Babai Jul 23 '18 at 9:15
• Sorry, I'm indicating the matrices as they are on your worksheet $$A=\left[\begin{matrix} 1&1\\0&1\end{matrix}\right]$$ and $$D=\left[\begin{matrix} 1&3\\0&2\end{matrix}\right]$$ – Davide Morgante Jul 23 '18 at 9:17
The text passage contains multiple mathematical questions related to determining if given pairs of matrices are conjugate in specific groups, and provides some discussion and hints towards solving these questions, but does not provide explicit answers.
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Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
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How to determine two matrices are conjugate.

Which of the following statements are true?

• The matrices $A=\left[ {\begin{array}{cc} 1 & 1 \\ 0 & 1\\ \end{array} } \right]$ and $B=\left[ {\begin{array}{cc} 1 & 0 \\ 1 & 1\\ \end{array} } \right]$ are conjugate in $GL_2(\mathbb{R})$
• The matrices $A=\left[ {\begin{array}{cc} 1 & 1 \\ 0 & 1\\ \end{array} } \right]$ and $B=\left[ {\begin{array}{cc} 1 & 0 \\ 1 & 1\\ \end{array} } \right]$ are conjugate in $SL_2(\mathbb{R})$
• The matrices $C=\left[ {\begin{array}{cc} 1 & 0 \\ 0 & 2\\ \end{array} } \right]$ and $D=\left[ {\begin{array}{cc} 1 & 3 \\ 0 & 2\\ \end{array} } \right]$ are conjugate in $GL_2(\mathbb{R})$

I know the conjugate matrices have the same eigenvalues. But all of this matrices have the same eigenvalues. I am not sure how to determine that which matrices are conjugate to each other.

The definition of conjugacy is: if $A$ and $B$ are two matrices such that $$A= P^{-1}BP$$ then they are conjugate. In $GL(2,\mathbb{R})$ is the same as similarity so not only the eigenvalues have to be preserved: mainly two similar matrices in $GL(2,\mathbb{R})$ have the same Jordan normal form
Notice that A is in Jordan canonical form (just look at the eigenvalues). Notice that the $D$ matrix is diagonalisable
Probabily this forum Why aren't this two matrices conjugate in $SL(2,\mathbb{R})$ can help for the second question.
• What is $D$ here? – Babai Jul 23 '18 at 9:15
• Sorry, I'm indicating the matrices as they are on your worksheet $$A=\left[\begin{matrix} 1&1\\0&1\end{matrix}\right]$$ and $$D=\left[\begin{matrix} 1&3\\0&2\end{matrix}\right]$$ – Davide Morgante Jul 23 '18 at 9:17
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The text passage contains multiple mathematical questions related to determining if given pairs of matrices are conjugate in specific groups, and provides some discussion and hints towards solving these questions, but does not provide explicit answers.

TYPE: MULTIPLE_QUESTIONS