document.write( "Question 1210438: Two matrices P and Q are
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Algebra.Com's Answer #852550 by ikleyn(53354)\"\" \"About 
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\n" ); document.write( "Two matrices P and Q are
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document.write( "Two matrices are given\r\n" );
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document.write( "    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\".\r\n" );
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document.write( "They are commutative under matrix multiplication.  It means P*Q = Q*P.\r\n" );
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document.write( "We have\r\n" );
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document.write( "    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\",\r\n" );
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document.write( "    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\".\r\n" );
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document.write( "The expressions in cells (1,1) and (2,2) are identical, so, they are not interested for us.\r\n" );
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document.write( "From cells (1,2), we have this equation\r\n" );
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document.write( "    6x^2 + 3x = 9 + 18x.\r\n" );
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document.write( "Cancel common factor 3\r\n" );
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document.write( "    2x^2 + x = 3 + 6x    (*)\r\n" );
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document.write( "    2x^2 - 5x - 3 = 0,\r\n" );
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document.write( "     (2x+1)*(x-3) = 0.\r\n" );
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document.write( "The roots are  -1/2  and  3.\r\n" );
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document.write( "From cells (2,1), we have this equation\r\n" );
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document.write( "    3+6x =  = 2x^2+x.\r\n" );
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document.write( "It is identical to equation (*), so, it does not carry any new information.\r\n" );
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document.write( "Now we select positive root x = 3.  It is the final answer:\r\n" );
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document.write( "    |      The problem has a unique answer x = 3        |\r\n" );
document.write( "    |              for positive 'x'.                    |\r\n" );
document.write( "    |   The matrices P and Q are commutative at x= 3.   |\r\n" );
document.write( "    +---------------------------------------------------+\r\n" );
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