document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1006566>Question 1006566</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If A is an invertible 2 x 2 matrix,then A^t is invertible and (At)^-1 = <BR>\n" );
document.write( "(A^-1)^t.<BR>\n" );
document.write( "           is this statement true or false and why? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #622675 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=622675\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Note: the notation \"[a,b;c,d]\" means \"the matrix with the first row being a,b and the second row being c,d\"\r<BR>\n" );
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document.write( "Matrix A is some 2 x 2 matrix, so let's set it up as<BR>\n" );
document.write( "A = [a,b;c,d]<BR>\n" );
document.write( "where a,b,c,d are constants\r<BR>\n" );
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document.write( "A is invertible ---> determinant D = ad-bc is nonzero\r<BR>\n" );
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document.write( "A^T = [a,c;b,d]<BR>\n" );
document.write( "determinant of A^T = ad-bc = determinant of matrix A\r<BR>\n" );
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document.write( "Note: only b and c swap places when we go from A to A^T. So both A and A^T have the same determinant\r<BR>\n" );
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document.write( "Since A has a nonzero determinant, so does A^T.\r<BR>\n" );
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document.write( "If A is invertible, then A^T is definitely invertible.\r<BR>\n" );
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document.write( "So the first part of the claim has been proven true.\r<BR>\n" );
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document.write( "Let's compute the inverse of matrix A\r<BR>\n" );
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document.write( "A = [a,b;c,d] <BR>\n" );
document.write( "A^(-1) = 1/D * [d,-b;-c,a] \r<BR>\n" );
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document.write( "Now let's transpose A^(-1) to get<BR>\n" );
document.write( "(A^(-1))^T = 1/D * [d,-c;-b,a] \r<BR>\n" );
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document.write( "Now compute the inverse of matrix A^T\r<BR>\n" );
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document.write( "A^T = [a,c;b,d]<BR>\n" );
document.write( "(A^T)^(-1) = 1/D * [d,-c;-b,a]\r<BR>\n" );
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document.write( "So,\r<BR>\n" );
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document.write( "(A^T)^(-1) = 1/D * [d,-c;-b,a]<BR>\n" );
document.write( "and<BR>\n" );
document.write( "(A^(-1))^T = 1/D * [d,-c;-b,a] \r<BR>\n" );
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document.write( "we can see that (A^T)^(-1) = (A^(-1))^T is true.\r<BR>\n" );
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document.write( "The second part of the statement has been proven true.\r<BR>\n" );
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document.write( "Both parts of the statement are true. A^T is invertible and the equation given holds true.<BR>\n" );
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