Matrices

Jan 2013
96
0
Find two 2x2 matrices with real entries A, B such that \(\displaystyle A^2+B^2=\left( \begin{array}{cc}2&3\\3&2\end{array}\right)\). (I need just an example.)
 

agentredlum

Math Team
Jul 2011
3,372
234
North America, 42nd parallel
yo79 said:
Find two 2x2 matrices with real entries A, B such that \(\displaystyle A^2+B^2=\left( \begin{array}{cc}2&3\\3&2\end{array}\right)\). (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.

\(\displaystyle \[\sqrt{2} \ \ \frac{3}{\sqrt{2}} \\ \\ \ 0 \ \ \ \ 0 \] \cdot \[\sqrt{2} \ \ \frac{3}{\sqrt{2}} \\ \\ \ 0 \ \ \ \ 0 \] \ + \ \[ 0 \ \ \ \ \ 0 \\ \\ \frac{3}{\sqrt{2}} \ \ \sqrt{2} \] \cdot \[ 0 \ \ \ \ \ 0 \\ \\ \frac{3}{\sqrt{2}} \ \ \sqrt{2} \] \ = \ \[2 \ \ 3 \\ 3 \ \ 2 \]\)

:D
 

mathbalarka

Math Team
Mar 2012
3,871
86
India, West Bengal
Nice question with a nice solution!
 

agentredlum

Math Team
Jul 2011
3,372
234
North America, 42nd parallel
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 \(\displaystyle \ p \ \ne \ 0\)

IF

\(\displaystyle A^2 \ + \ B^2 \ = \ \[p \ q \\ q \ p \]\)

THEN a solution will be

\(\displaystyle A \ = \[ \sqrt{p} \ \ \frac{q}{ \sqrt{p}} \\ \ 0 \ \ \ \ 0 \]\)

\(\displaystyle 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 \(\displaystyle \ p \ = \ 0 \ \ \text{AND} \ \ q \ \ne \ 0 \\)?'

A place to start could be ,

Find a 2×2 matrix A such that

\(\displaystyle A^2 \ = \ \[ 0 \ 1 \\ 1 \ 0 \]\)

Can it be found ?

:D
 
Jan 2013
96
0
Of course not, if A has real entries ( det(A)^2=-1)!
 

CRGreathouse

Forum Staff
Nov 2006
16,046
936
UTC -5
agentredlum said:
Find a 2×2 matrix A such that

\(\displaystyle 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.
 

agentredlum

Math Team
Jul 2011
3,372
234
North America, 42nd parallel
Can we find A with complex entries?

:D
 

agentredlum

Math Team
Jul 2011
3,372
234
North America, 42nd parallel
Find 4 distinct matrices A such that

\(\displaystyle A^2 \ = \ \[4 \ 0 \\ 0 \ 4 \]\)

:D