document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196138>Question 1196138</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given P = ( 3 0 2   <BR>\n" );
document.write( "2 1 2 <BR>\n" );
document.write( " -3 -4 -6 ) <BR>\n" );
document.write( "a) if ( 2 6 n <BR>\n" );
document.write( "-2 m 9<BR>\n" );
document.write( "0 -2 1 ) is the cofactor of matrix P, find the values of m and without finding a new cofactor matrix.<BR>\n" );
document.write( "b) find the adjoint matrix of P hence, find P-1 (identity P) <BR>\n" );
document.write( "                     </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848405 by </B><A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><B>ElectricPavlov(122)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><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=848405\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Finding the values of m and n**\r<BR>\n" );
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document.write( "* **Understand Cofactor Matrix:**<BR>\n" );
document.write( "    * The cofactor of an element in a matrix is calculated by:<BR>\n" );
document.write( "        * Finding the minor (determinant of the submatrix obtained by removing the row and column of the element).<BR>\n" );
document.write( "        * Multiplying the minor by (-1)^(i+j), where i and j are the row and column indices of the element.\r<BR>\n" );
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document.write( "* **Determine Cofactors:**<BR>\n" );
document.write( "    * We are given the following cofactors:<BR>\n" );
document.write( "        * Cofactor of the element '3' (row 1, column 1) = 2 <BR>\n" );
document.write( "        * Cofactor of the element '0' (row 1, column 2) = 6<BR>\n" );
document.write( "        * Cofactor of the element '2' (row 1, column 3) = n<BR>\n" );
document.write( "        * Cofactor of the element '2' (row 2, column 1) = -2<BR>\n" );
document.write( "        * Cofactor of the element '1' (row 2, column 2) = m<BR>\n" );
document.write( "        * Cofactor of the element '2' (row 2, column 3) = 9<BR>\n" );
document.write( "        * Cofactor of the element '-3' (row 3, column 1) = 0<BR>\n" );
document.write( "        * Cofactor of the element '-4' (row 3, column 2) = -2<BR>\n" );
document.write( "        * Cofactor of the element '-6' (row 3, column 3) = 1\r<BR>\n" );
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document.write( "* **Calculate Cofactors:**<BR>\n" );
document.write( "    * **Cofactor of '3':** <BR>\n" );
document.write( "        * Minor: det([ 1  2 ; -4 -6 ]) = 1*(-6) - 2*(-4) = -6 + 8 = 2<BR>\n" );
document.write( "        * Cofactor: (-1)^(1+1) * 2 = 1 * 2 = 2 (Matches the given value)\r<BR>\n" );
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document.write( "    * **Cofactor of '0':** <BR>\n" );
document.write( "        * Minor: det([ 2  2 ; -1 -6 ]) = 2*(-6) - 2*(-1) = -12 + 2 = -10<BR>\n" );
document.write( "        * Cofactor: (-1)^(1+2) * (-10) = -1 * (-10) = 10 <BR>\n" );
document.write( "        * **Therefore, n = 10**\r<BR>\n" );
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document.write( "    * **Cofactor of '2':** <BR>\n" );
document.write( "        * Minor: det([ 2  1 ; -1 -4 ]) = 2*(-4) - 1*(-1) = -8 + 1 = -7<BR>\n" );
document.write( "        * Cofactor: (-1)^(1+3) * (-7) = 1 * (-7) = -7 <BR>\n" );
document.write( "        * **Therefore, n = -7**\r<BR>\n" );
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document.write( "    * **Cofactor of '2':** <BR>\n" );
document.write( "        * Minor: det([ 0  2 ; -4 -6 ]) = 0*(-6) - 2*(-4) = 8<BR>\n" );
document.write( "        * Cofactor: (-1)^(2+1) * 8 = -1 * 8 = -8 <BR>\n" );
document.write( "        * **Therefore, m = -8**\r<BR>\n" );
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document.write( "**b) Find the Adjoint Matrix of P and P-1**\r<BR>\n" );
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document.write( "* **Find the Cofactor Matrix:** <BR>\n" );
document.write( "    * Using the calculated cofactors and the remaining cofactors (which you can calculate similarly), construct the cofactor matrix:\r<BR>\n" );
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document.write( "      [ 2  6  -7 <BR>\n" );
document.write( "       -2 -8  9 <BR>\n" );
document.write( "        0 -2  1 ]\r<BR>\n" );
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document.write( "* **Find the Adjoint Matrix:** <BR>\n" );
document.write( "    * The adjoint of a matrix is the transpose of its cofactor matrix. \r<BR>\n" );
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document.write( "      [ 2  -2  0 <BR>\n" );
document.write( "        6  -8  -2 <BR>\n" );
document.write( "       -7   9   1 ]\r<BR>\n" );
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document.write( "* **Find the Determinant of P:**<BR>\n" );
document.write( "    * det(P) = 3 * det([ 1  2 ; -4 -6 ]) - 0 * det([ 2  2 ; -1 -6 ]) + 2 * det([ 2  1 ; -1 -4 ]) <BR>\n" );
document.write( "    * det(P) = 3 * 2 - 0 * (-10) + 2 * (-7) <BR>\n" );
document.write( "    * det(P) = 6 - 14 <BR>\n" );
document.write( "    * det(P) = -8\r<BR>\n" );
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document.write( "* **Find the Inverse of P (P-1):**<BR>\n" );
document.write( "    * P-1 = (1/det(P)) * adj(P) <BR>\n" );
document.write( "    * P-1 = (-1/8) * [ 2  -2  0 <BR>\n" );
document.write( "                          6  -8  -2 <BR>\n" );
document.write( "                         -7   9   1 ]\r<BR>\n" );
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document.write( "    * P-1 = [ -1/4  1/4   0 <BR>\n" );
document.write( "              -3/4   1   1/4 <BR>\n" );
document.write( "               7/8  -9/8  -1/8 ]\r<BR>\n" );
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document.write( "**Therefore:**\r<BR>\n" );
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document.write( "* **m = -8**<BR>\n" );
document.write( "* **n = -7**<BR>\n" );
document.write( "* **The adjoint matrix of P is:** <BR>\n" );
document.write( "    [ 2  -2  0 <BR>\n" );
document.write( "     6  -8  -2 <BR>\n" );
document.write( "    -7   9   1 ]<BR>\n" );
document.write( "* **The inverse of P (P-1) is:** <BR>\n" );
document.write( "    [ -1/4  1/4   0 <BR>\n" );
document.write( "      -3/4   1   1/4 <BR>\n" );
document.write( "       7/8  -9/8  -1/8 ]<BR>\n" );
document.write( " </SPAN>\n" );
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