document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=35533>Question 35533</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove that if &#955; is an eigencalue of an invertible matrix A and x is a corresponding eigenvector, then 1/&#955; is an eigenvalue of A inverese (A(-1)) , and x is a corresponding eigenvector </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #21679 by </B><A HREF=/tutors/aboutme.mpl?userid=narayaba><B>narayaba(40)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=narayaba><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=21679\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given that &#955; is an eigenvalue of the invertibe matrix with x as its eigen vector.<BR>\n" );
document.write( "This means Ax = &#955;x such that x is non-zero\r<BR>\n" );
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document.write( "Ax = &#955;x\r<BR>\n" );
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document.write( "lets multiply both side of the above equation by the inverse of A( A^-1) from the left. This is possibe since the inverse of A exits according to the problem definition.\r<BR>\n" );
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document.write( "(A^-1)*A*x = (A^-1)*&#955;x\r<BR>\n" );
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document.write( "since (A^-1)*A  = I(identity matrix)\r<BR>\n" );
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document.write( "(A^-1)*&#955;x = x\r<BR>\n" );
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document.write( "muliply by 1/&#955; both sides<BR>\n" );
document.write( "(A^-1)*x = 1/&#955;x\r<BR>\n" );
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document.write( "this shows that 1/&#955; is eigenvalue of matrix A^-1 with eigen vector x<BR>\n" );
document.write( " </SPAN>\n" );
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