document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206684>Question 1206684</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello, I need help with the following problem:\r<BR>\n" );
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document.write( "Three basis are given in the plane. With respect to those basis a point has as components (x1,x2), (y1,y2) and (z1,z2). Suppose that [y1 y2] = [x1 x2]A, [z1 z2] = [x1 x2]B and [z1 z2] = [y1 y2]C being A,B,C 2x2 matrices. Express C as a function of A and B.\r<BR>\n" );
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document.write( "I very much appreciate your help. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847960 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=847960\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We're given the following relationships:\r<BR>\n" );
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document.write( "1.  [y1 y2] = [x1 x2]A<BR>\n" );
document.write( "2.  [z1 z2] = [x1 x2]B<BR>\n" );
document.write( "3.  [z1 z2] = [y1 y2]C\r<BR>\n" );
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document.write( "We want to express C as a function of A and B. \r<BR>\n" );
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document.write( "**Step 1: Substitute [y1 y2] in equation 3:**\r<BR>\n" );
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document.write( "From equation 1, we know that [y1 y2] = [x1 x2]A. Substitute this into equation 3:\r<BR>\n" );
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document.write( "[z1 z2] = ([x1 x2]A)C\r<BR>\n" );
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document.write( "**Step 2: Substitute [z1 z2] from equation 2:**\r<BR>\n" );
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document.write( "From equation 2, we know that [z1 z2] = [x1 x2]B. Substitute this into the previous equation:\r<BR>\n" );
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document.write( "[x1 x2]B = ([x1 x2]A)C\r<BR>\n" );
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document.write( "**Step 3: Simplify and Solve for C:**\r<BR>\n" );
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document.write( "Since [x1 x2] is a common factor on both sides, we can cancel it out:\r<BR>\n" );
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document.write( "B = AC\r<BR>\n" );
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document.write( "**Now, to isolate C, we can multiply both sides by the inverse of A (assuming A is invertible):**\r<BR>\n" );
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document.write( "A⁻¹B = C\r<BR>\n" );
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document.write( "**Therefore, C = A⁻¹B.**<BR>\n" );
document.write( " </SPAN>\n" );
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