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 January 10th, 2014, 07:55 AM #1 Member   Joined: Jan 2013 Posts: 96 Thanks: 0 Matrices Find two 2x2 matrices with real entries A, B such that $A^2+B^2=\left( \begin{array}{cc}2&3\\3=&2\end{array}\right)=$. (I need just an example.)
January 10th, 2014, 09:32 PM   #2
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Re: Matrices

Quote:
 Originally Posted by yo79 Find two 2x2 matrices with real entries A, B such that $A^2+B^2=\left( \begin{array}{cc}2&3\\3=&2\end{array}\right)=$. (I need just an example.)
Nice question!

Well i played around with it for a while till i realized we could work with rows as one way to get an answer.

$$\sqrt{2} \ \ \frac{3}{\sqrt{2}} \\ \\ \ 0 \ \ \ \ 0$ \cdot $\sqrt{2} \ \ \frac{3}{\sqrt{2}} \\ \\ \ 0 \ \ \ \ 0$ \ + \ $0 \ \ \ \ \ 0 \\ \\ \frac{3}{\sqrt{2}} \ \ \sqrt{2}$ \cdot $0 \ \ \ \ \ 0 \\ \\ \frac{3}{\sqrt{2}} \ \ \sqrt{2}$ \= \ $2 \ \ 3 \\ 3 \ \ 2$$

 January 11th, 2014, 08:10 AM #3 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory Re: Matrices Nice question with a nice solution!
 January 11th, 2014, 10:22 AM #4 Member   Joined: Jan 2013 Posts: 96 Thanks: 0 Re: Matrices Thank you!
 January 12th, 2014, 03:06 AM #5 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 Re: Matrices You're welcome. Just in case you haven't noticed , we now have an easy solution to an infinite amount of problems of this form. For $\ p \ \ne \ 0$ IF $A^2 \ + \ B^2 \= \ $p \ q \\ q \ p$$ THEN a solution will be $A \= $\sqrt{p} \ \ \frac{q}{ \sqrt{p}} \\ \ 0 \ \ \ \ 0$$ $B \= \ $\ \ 0 \ \ \ \ 0 \\ \frac{q}{ \sqrt{p}} \ \ \sqrt{p}$$ We can also transpose both A , B and get another family of solutions. Then we would be working with columns. An interesting question to ask is 'what if $\ p \= \ 0 \ \ \text{AND} \ \ q \ \ne \ 0 $/extract_itex]?' A place to start could be , Find a 2×2 matrix A such that $A^2 \= \ \[ 0 \ 1 \\ 1 \ 0$$ Can it be found ?
 January 12th, 2014, 08:12 AM #6 Member   Joined: Jan 2013 Posts: 96 Thanks: 0 Re: Matrices Of course not, if A has real entries ( det(A)^2=-1)!
January 12th, 2014, 08:39 AM   #7
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Re: Matrices

Quote:
 Originally Posted by agentredlum Find a 2×2 matrix A such that $A^2 \= \ $0 \ 1 \\ 1 \ 0$$ Can it be found ?
Not in the real numbers. There is a unique complex solution up to the order of the variables.

 January 12th, 2014, 08:42 AM #8 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 Re: Matrices Can we find A with complex entries?
January 12th, 2014, 11:06 AM   #9
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Re: Matrices

Quote:
 Originally Posted by CRGreathouse Not in the real numbers.
Indeed.

 January 13th, 2014, 02:44 AM #10 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 Re: Matrices Find 4 distinct matrices A such that $A^2 \= \ $4 \ 0 \\ 0 \ 4$$

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