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)   (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 =   and  Q = .


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

We have

    P*Q =  *  = ,


    Q*P =  * ( = .


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)   (Show Source): You can put this solution on YOUR website!






The entry in row 2 column 1 of matrix PQ is

The entry in row 2 column 1 of matrix QP is

those entries are the same, so





or

The positive solution is

ANSWER: x=3
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