Question 1210438
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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. 
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<pre>
Two matrices are given

    P = {{{(matrix(2,2,  x^2, 3,   1, 3x))}}}  and  Q = {{{(matrix(2,2,  3, 6,  2, x))}}}.


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

We have

    P*Q = {{{(matrix(2,2,  x^2, 3,   1, 3x))}}} * {{{(matrix(2,2,  3, 6,  2, x))}}} = {{{(matrix(2,2, 3x^2+6, 6x^2+3x, 3+6x, 6+3x^2))}}},


    Q*P = {{{(matrix(2,2,  3, 6,  2, x))}}} * ({{{(matrix(2,2,  x^2, 3,   1, 3x))}}} = {{{(matrix(2,2, 3x^2+6, 9+18x, 2x^2+x, 6+3x^2))}}}.


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.   |
    +---------------------------------------------------+
</pre>

Solved.