document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196231>Question 1196231</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If m= 1   -1    k<BR>\n" );
document.write( "          4    7    3<BR>\n" );
document.write( "          -1  12   -2<BR>\n" );
document.write( "Evaluate in term of k the determinant of m. Hence, if x= x<BR>\n" );
document.write( "                                                                                                   y<BR>\n" );
document.write( "                                                                                                   z<BR>\n" );
document.write( "slove the equation MX=  1<BR>\n" );
document.write( "                                            11<BR>\n" );
document.write( "                                            21<BR>\n" );
document.write( "when k=2, by the cot method </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848403 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=848403\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Evaluate the Determinant of Matrix 'm'**\r<BR>\n" );
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document.write( "* **Expand along the first row:**\r<BR>\n" );
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document.write( "   det(m) = 1 * det( <BR>\n" );
document.write( "                       7   3<BR>\n" );
document.write( "                       12 -2 <BR>\n" );
document.write( "                    ) <BR>\n" );
document.write( "           - (-1) * det( <BR>\n" );
document.write( "                           4   3<BR>\n" );
document.write( "                           -1 -2 <BR>\n" );
document.write( "                       ) <BR>\n" );
document.write( "           + K * det( <BR>\n" );
document.write( "                           4   7<BR>\n" );
document.write( "                           -1  12 <BR>\n" );
document.write( "                       ) \r<BR>\n" );
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document.write( "   det(m) = 1 * (7 * (-2) - 3 * 12) + 1 * (4 * (-2) - 3 * (-1)) + K * (4 * 12 - 7 * (-1))<BR>\n" );
document.write( "   det(m) = 1 * (-14 - 36) + 1 * (-8 + 3) + K * (48 + 7)<BR>\n" );
document.write( "   det(m) = -50 - 5 + 55K<BR>\n" );
document.write( "   det(m) = 55K - 55\r<BR>\n" );
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document.write( "**2. Solve the Equation 'mx = (1, 11, 21)' when K = 2**\r<BR>\n" );
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document.write( "* **Substitute K = 2 into matrix 'm':**\r<BR>\n" );
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document.write( "   m = ( <BR>\n" );
document.write( "         1  -1   2<BR>\n" );
document.write( "         4   7   3<BR>\n" );
document.write( "         -1  12 -2 <BR>\n" );
document.write( "      )\r<BR>\n" );
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document.write( "* **Set up the augmented matrix:**\r<BR>\n" );
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document.write( "   [ <BR>\n" );
document.write( "      1  -1   2  |  1<BR>\n" );
document.write( "      4   7   3  |  11<BR>\n" );
document.write( "      -1  12 -2  |  21<BR>\n" );
document.write( "   ]\r<BR>\n" );
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document.write( "* **Perform row operations to get the matrix in row-echelon form:**\r<BR>\n" );
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document.write( "   1. **R2 = R2 - 4R1** <BR>\n" );
document.write( "      [ <BR>\n" );
document.write( "         1  -1   2  |  1<BR>\n" );
document.write( "         0  11  -5  |  7<BR>\n" );
document.write( "         -1  12 -2  |  21<BR>\n" );
document.write( "      ]\r<BR>\n" );
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document.write( "   2. **R3 = R3 + R1** <BR>\n" );
document.write( "      [ <BR>\n" );
document.write( "         1  -1   2  |  1<BR>\n" );
document.write( "         0  11  -5  |  7<BR>\n" );
document.write( "         0  11   0  |  22<BR>\n" );
document.write( "      ]\r<BR>\n" );
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document.write( "   3. **R3 = R3 - R2** <BR>\n" );
document.write( "      [ <BR>\n" );
document.write( "         1  -1   2  |  1<BR>\n" );
document.write( "         0  11  -5  |  7<BR>\n" );
document.write( "         0   0   5  |  15<BR>\n" );
document.write( "      ]\r<BR>\n" );
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document.write( "* **Solve for Z:**\r<BR>\n" );
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document.write( "   * 5Z = 15<BR>\n" );
document.write( "   * Z = 3\r<BR>\n" );
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document.write( "* **Back-substitute to find Y:**\r<BR>\n" );
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document.write( "   * 11Y - 5Z = 7<BR>\n" );
document.write( "   * 11Y - 5(3) = 7<BR>\n" );
document.write( "   * 11Y - 15 = 7<BR>\n" );
document.write( "   * 11Y = 22<BR>\n" );
document.write( "   * Y = 2\r<BR>\n" );
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document.write( "* **Back-substitute to find X:**\r<BR>\n" );
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document.write( "   * X - Y + 2Z = 1<BR>\n" );
document.write( "   * X - 2 + 2(3) = 1<BR>\n" );
document.write( "   * X - 2 + 6 = 1<BR>\n" );
document.write( "   * X = -3\r<BR>\n" );
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document.write( "**Therefore, when K = 2, the solution to the equation 'mx = (1, 11, 21)' is:**\r<BR>\n" );
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document.write( "* **X = -3**<BR>\n" );
document.write( "* **Y = 2**<BR>\n" );
document.write( "* **Z = 3** \r<BR>\n" );
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document.write( "I hope this comprehensive solution is helpful!<BR>\n" );
document.write( " </SPAN>\n" );
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