Algebra.Com's Answer #677365 by ikleyn(52790)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
.
\n" );
document.write( "\r\n" );
document.write( "Calculate the determinant of the matrix A =  .\r\n" );
document.write( "\r\n" );
document.write( "If you are unfamiliar with it or want to refresh your knowledge, see the lesson\r\n" );
document.write( "\r\n" );
document.write( " - Determinant of a 3x3 matrix \r\n" );
document.write( "\r\n" );
document.write( "in this site.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The determinant is equal to  =  .\r\n" );
document.write( "\r\n" );
document.write( "The general theory says:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " If the determinant is not zero, then the system has a unique solution.\r\n" );
document.write( "\r\n" );
document.write( " The degenerate cases are those when the determinant is zero:\r\n" );
document.write( "\r\n" );
document.write( " d(k) =  = 0.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "One root of this equation is k = 1.\r\n" );
document.write( "\r\n" );
document.write( "Hence, the polynomial is multiple of (k-1), and the ratio  is  .\r\n" );
document.write( "\r\n" );
document.write( "Then the two other roots of the determinant are the zeroes of this quadratic polynomial  . \r\n" );
document.write( "\r\n" );
document.write( "You can easily find these zeroes using the quadratic formula:\r\n" );
document.write( "\r\n" );
document.write( " =  and =  .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Next, if k = 1, your original system consists of three identical equations.\r\n" );
document.write( "Hence, the system has infinitely many solutions at k = 1.\r\n" );
document.write( "\r\n" );
document.write( "If k = or k =  , then the system has infinitely many solutions, too.\r\n" );
document.write( "\r\n" );
document.write( " Indeed, if you add all three equations at these values of k, you will get the equation \r\n" );
document.write( "\r\n" );
document.write( " (k+2)x + (k+2)y + (k+2)z = 3, or\r\n" );
document.write( "\r\n" );
document.write( " x + y + z =  . (*)\r\n" );
document.write( "\r\n" );
document.write( " Now distract the equation (1) from (*), and you will find the solution for x.\r\n" );
document.write( " Distract then the equation (2) from (*), and you will find the solution for y.\r\n" );
document.write( " Finally, distract the equation (3) from (*), and you will find the solution for z.\r\n" );
document.write( "\r\n" );
document.write( " Thus the system has at least one solution for k = and/or k = , having the determinant equal to zero at these values of k.\r\n" );
document.write( " It is enough for the system to have infinitely many solutions then.\r\n" );
document.write( " \r
\n" );
document.write( "\n" );
document.write( "Answer. The system has infinitely many solutions at k = 1, k = and k = .\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( " The system has a unique solution for all other values of k.\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "------------------------------
\n" );
document.write( "The useful references on this theme are these lessons\r
\n" );
document.write( "\n" );
document.write( " - Determinant of a 3x3 matrix \r
\n" );
document.write( "\n" );
document.write( " - HOW TO solve system of linear equations in three unknowns using determinant (Cramer's rule)\r
\n" );
document.write( "\n" );
document.write( " - Solving systems of linear equations in three unknowns using determinant (Cramer's rule)\r
\n" );
document.write( "\n" );
document.write( "in this site.\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "Also, you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" );
document.write( "\n" );
document.write( " - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "The referred lessons are the part of this online textbook under the topic
\n" );
document.write( " \"3x3-Matrices, determinants, Cramer's rule for systems in three unknowns\" \r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( " \n" );
document.write( " |