document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201590>Question 1201590</A>: <I><SPAN STYLE=\"word-break: break-all;\"> prove that, for square matrices A and B, AB=BA if and only if(A-B)(A+B)=A²-B².<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #836044 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=836044\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The template P if and only if Q breaks into two pieces<ul><li>If P, then Q</li><li>If Q, then P</li></ul><BR>\n" );
document.write( "For this problem,<BR>\n" );
document.write( "AB = BA if and only if (A-B)(A+B) = A^2-B^2<BR>\n" );
document.write( "breaks into<ul><li>If AB = BA, then (A-B)(A+B) = A^2-B^2</li><li>If (A-B)(A+B) = A^2-B^2, then AB = BA</li></ul>\r<BR>\n" );
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document.write( "Part 1)<BR>\n" );
document.write( "If AB = BA, then (A-B)(A+B) = A^2-B^2\r<BR>\n" );
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document.write( "We'll start with (A-B)(A+B) and try to reach A^2-B^2 through use of AB = BA\r<BR>\n" );
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document.write( "(A-B)(A+B) = A(A+B)-B(A+B)<BR>\n" );
document.write( "(A-B)(A+B) = (A^2+AB)+(-BA-B^2)<BR>\n" );
document.write( "(A-B)(A+B) = (A^2+AB)+(-AB-B^2) ... use AB = BA<BR>\n" );
document.write( "(A-B)(A+B) = A^2+(AB-AB)-B^2<BR>\n" );
document.write( "(A-B)(A+B) = A^2+0*AB-B^2<BR>\n" );
document.write( "(A-B)(A+B) = A^2+0-B^2<BR>\n" );
document.write( "(A-B)(A+B) = A^2-B^2\r<BR>\n" );
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document.write( "We have proven that if AB=BA, then (A-B)(A+B) leads to A^2-B^2\r<BR>\n" );
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document.write( "In other words, if AB=BA, then (A-B)(A+B) = A^2-B^2\r<BR>\n" );
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document.write( "Part 2)\r<BR>\n" );
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document.write( "If (A-B)(A+B) = A^2-B^2, then AB = BA\r<BR>\n" );
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document.write( "(A-B)(A+B) = A^2-B^2<BR>\n" );
document.write( "A(A+B)-B(A+B) = A^2-B^2<BR>\n" );
document.write( "(A^2+AB)+(-BA-B^2) = A^2-B^2<BR>\n" );
document.write( "A^2+(AB-BA)-B^2 = A^2-B^2<BR>\n" );
document.write( "(AB-BA)-B^2 = -B^2<BR>\n" );
document.write( "AB-BA = 0<BR>\n" );
document.write( "AB = BA\r<BR>\n" );
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document.write( "The second portion of the \"if and only if\" statement has been confirmed.\r<BR>\n" );
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document.write( "Therefore, overall we can say AB = BA if and only if (A-B)(A+B) = A^2-B^2<BR>\n" );
document.write( "Matrices A and B must be square, and the same size.<BR>\n" );
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