document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1137268>Question 1137268</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the inverse of matrix of elements cos theta - sin theta                 <BR>\n" );
document.write( "                                       sin theta   cos theta </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #755152 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=755152\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"times\" color=\"black\" size=\"3\"><BR>\n" );
document.write( "Check out <a  rel=nofollow HREF=\"http://www.mathcentre.ac.uk/resources/uploaded/sigma-matrices7-2009-1.pdf\">this article</a> to see how to find the inverse of a 2x2 matrix. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Part of the steps will have us compute the determinant. The determinant of the 2x2 matrix is found through this formula<BR>\n" );
document.write( "E = determinant<BR>\n" );
document.write( "E = a*d - b*c<BR>\n" );
document.write( "where a,b,c,d are arranged like this in the matrix<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \begin{bmatrix}a & b\\c & d\end{bmatrix}\"><BR>\n" );
document.write( "In this case,<BR>\n" );
document.write( "a = cos(theta)<BR>\n" );
document.write( "b = -sin(theta)<BR>\n" );
document.write( "c = sin(theta)<BR>\n" );
document.write( "d = cos(theta)<BR>\n" );
document.write( "which we can see through this comparison<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \begin{bmatrix}a & b\\c & d\end{bmatrix} = \begin{bmatrix}\cos(\theta) & -\sin(\theta)\\\sin(\theta) & \cos(\theta)\end{bmatrix}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the determinant is,<BR>\n" );
document.write( "E = a*d - b*c<BR>\n" );
document.write( "E = cos(theta)*cos(theta) - (-sin(theta))*sin(theta)<BR>\n" );
document.write( "E = cos(theta)*cos(theta) + sin(theta)*sin(theta)<BR>\n" );
document.write( "E = cos^2(theta) + sin^2(theta)<BR>\n" );
document.write( "E = 1<BR>\n" );
document.write( "For that last step, you use the pythagorean trig identity. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Therefore, the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28a%2Ad+-+b%2Ac%29 ALIGN=MIDDLE  ALT=\"1%2F%28a%2Ad+-+b%2Ac%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equal to 1 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28a%2Ad+-+b%2Ac%29+=+1%2F1+=+1 ALIGN=MIDDLE  ALT=\"1%2F%28a%2Ad+-+b%2Ac%29+=+1%2F1+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, meaning that this portion does not affect the result. This is because multiplying by 1 does not change the answer. We can ignore this piece because the determinant is 1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In that same article I posted, note how the original matrix looks like this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \begin{bmatrix}a & b\\c & d\end{bmatrix}\"><BR>\n" );
document.write( "and it becomes this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \begin{bmatrix}d & -b\\-c & a\end{bmatrix}\"><BR>\n" );
document.write( "You swap a and d; also change the signs of b and c but keep them in the same spot.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since <BR>\n" );
document.write( "a = cos(theta)<BR>\n" );
document.write( "b = -sin(theta)<BR>\n" );
document.write( "c = sin(theta)<BR>\n" );
document.write( "d = cos(theta)<BR>\n" );
document.write( "we can say...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \begin{bmatrix}d & -b\\-c & a\end{bmatrix} = \begin{bmatrix}\cos(\theta) & -(-\sin(\theta))\\-\sin(\theta) & \cos(\theta)\end{bmatrix}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \begin{bmatrix}d & -b\\-c & a\end{bmatrix} = \begin{bmatrix}\cos(\theta) & \sin(\theta)\\-\sin(\theta) & \cos(\theta)\end{bmatrix}\"><BR>\n" );
document.write( "which is the inverse of the original matrix. I'll let you confirm this by computing A*B and B*A, where A is the original matrix and B is the inverse matrix. You should get AB = I and BA = I. Recall that the matrix I is the identity matrix.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );