SOLUTION: Two matrices P and Q are ┌x² 3┐ and ┌3 6┐respectively. │1 3x│ │2 x│ Given that P and Q are commutative under matrix multiplication. Find the positi

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Question 1210438: Two matrices P and Q are
┌x² 3┐ and ┌3 6┐respectively.
│1 3x│ │2 x│
Given that P and Q are commutative under matrix multiplication. Find the positive value of x.
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(53354) About Me  (Show Source):
You can put this solution on YOUR website!
.
Two matrices P and Q are
┌x² 3┐ and ┌3 6┐respectively.
│1 3x│ │2 x│
Given that P and Q are commutative under matrix multiplication. Find the positive value of x.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

Two matrices are given

    P = %28matrix%282%2C2%2C++x%5E2%2C+3%2C+++1%2C+3x%29%29  and  Q = %28matrix%282%2C2%2C++3%2C+6%2C++2%2C+x%29%29.


They are commutative under matrix multiplication.  It means P*Q = Q*P.

We have

    P*Q = %28matrix%282%2C2%2C++x%5E2%2C+3%2C+++1%2C+3x%29%29 * %28matrix%282%2C2%2C++3%2C+6%2C++2%2C+x%29%29 = %28matrix%282%2C2%2C+3x%5E2%2B6%2C+6x%5E2%2B3x%2C+3%2B6x%2C+6%2B3x%5E2%29%29,


    Q*P = %28matrix%282%2C2%2C++3%2C+6%2C++2%2C+x%29%29 * (%28matrix%282%2C2%2C++x%5E2%2C+3%2C+++1%2C+3x%29%29 = %28matrix%282%2C2%2C+3x%5E2%2B6%2C+9%2B18x%2C+2x%5E2%2Bx%2C+6%2B3x%5E2%29%29.


The expressions in cells (1,1) and (2,2) are identical, so, they are not interested for us.


From cells (1,2), we have this equation

    6x^2 + 3x = 9 + 18x.


Cancel common factor 3

    2x^2 + x = 3 + 6x    (*)

    2x^2 - 5x - 3 = 0,

     (2x+1)*(x-3) = 0.


The roots are  -1/2  and  3.


From cells (2,1), we have this equation

    3+6x =  = 2x^2+x.


It is identical to equation (*), so, it does not carry any new information.


Now we select positive root x = 3.  It is the final answer:

    +---------------------------------------------------+
    |      The problem has a unique answer x = 3        |
    |              for positive 'x'.                    |
    |   The matrices P and Q are commutative at x= 3.   |
    +---------------------------------------------------+

Solved.



Answer by greenestamps(13250) About Me  (Show Source):
You can put this solution on YOUR website!


P=matrix%282%2C2%2Cx%5E2%2C3%2C1%2C3x%29

Q=matrix%282%2C2%2C3%2C6%2C2%2Cx%29

The entry in row 2 column 1 of matrix PQ is %281%29%283%29%2B%283x%29%282%29=3%2B6x

The entry in row 2 column 1 of matrix QP is %282%29%28x%5E2%29%2B%28x%29%281%29=2x%5E2%2Bx

those entries are the same, so

3%2B6x=2x%5E2%2Bx
2x%5E2-5x-3=0
%282x%2B1%29%28x-3%29=0

x=-1%2F2 or x=3

The positive solution is

ANSWER: x=3