document.write( "Question 354999: Please help me solve this equation: Solve the following system of equations by determining the inverse of the matrix of coefficients and then using matrix multiplication.\r
\n" ); document.write( "\n" ); document.write( " x+3y+z = 4
\n" ); document.write( " 2x+2y+z = -1
\n" ); document.write( " 2x+3y+z = 3 \n" ); document.write( "
Algebra.Com's Answer #253592 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
\"system%28x%2B3y%2Bz+=+4%2C%0D%0A2x%2B2y%2Bz+=+-1%2C%0D%0A2x%2B3y%2Bz+=+3%29\"
\n" ); document.write( "
\r\n" );
document.write( "I will assume you already know how to find the inverse\r\n" );
document.write( "of a matrix, and how to multiply two matrices. If you don't, \r\n" );
document.write( "post again asking how.  Each of those is is a separate topic.\r\n" );
document.write( "\r\n" );
document.write( "First we form three matrices, A, X, and B.\r\n" );
document.write( "\r\n" );
document.write( "1. Matrix A is the 3x3 coefficient matrix A, which consists \r\n" );
document.write( "of just the three columns of x, y, and z coefficients. in \r\n" );
document.write( "that order, but does not contain the column of constants.\r\n" );
document.write( "\r\n" );
document.write( "\"A=%28matrix%283%2C3%2C1%2C3%2C1%2C2%2C2%2C1%2C2%2C3%2C1%29%29\". \r\n" );
document.write( "\r\n" );
document.write( "2. Matrix X is the 3x1 matrix of variables \"X=%28matrix%283%2C1%2Cx%2Cy%2Cz%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "3. Matrix B is the 3x1 matrix, whose only column is the\r\n" );
document.write( "column of constants: \"%28matrix%283%2C1%2C4%2C-1%2C3%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "Next we form the matrix equation:\r\n" );
document.write( "\r\n" );
document.write( "       \"A%2AX+=+B\"\r\n" );
document.write( "\r\n" );
document.write( "or\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "To solve the equation\r\n" );
document.write( "\r\n" );
document.write( "       \"A%2AX+=+B\"\r\n" );
document.write( "\r\n" );
document.write( "we left-multiply both sides by \"A%5E%28-1%29\", the inverse of \"A\".\r\n" );
document.write( "\r\n" );
document.write( " \"A%5E%28-1%29%2A%28A%2AX%29+=+A%5E%28-1%29%2AB\"\r\n" );
document.write( "\r\n" );
document.write( "Then since the associatitive principle holds for matrix multiplication,\r\n" );
document.write( "(even though the commutative principle DOES NOT!!!), we can move\r\n" );
document.write( "the parentheses on the left around the first two matrix factors:\r\n" );
document.write( "\r\n" );
document.write( "\"%28A%5E%28-1%29%2AA%29%2AX+=+A%5E%28-1%29%2AB\"\r\n" );
document.write( "\r\n" );
document.write( "Now since \"%28A%5E%28-1%29%2AA%29=I\", where I is the identity matrix, the\r\n" );
document.write( "above becomes:\r\n" );
document.write( "\r\n" );
document.write( "\"I%2AX+=+A%5E%28-1%29%2AB\"\r\n" );
document.write( "\r\n" );
document.write( "and by the identity property:\r\n" );
document.write( "\r\n" );
document.write( "\"X=A%5E%28-1%29%2AB\"\r\n" );
document.write( "\r\n" );
document.write( "Performing these operations with the actual matrices we have\r\n" );
document.write( "the equation \"AX=B\"\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Next we form the inverse of A, which is written A-1.\r\n" );
document.write( "\r\n" );
document.write( "\"A%5E%28-1%29=%28matrix%283%2C3%2C-1%2C+0%2C+1%2C+0%2C+-1%2C+1%2C+2%2C+3%2C+-4%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "Remember I assume you know where I got this inverse.  It is a whole separate\r\n" );
document.write( "problem on how to find it.  If you don't know how, post again asking how.\r\n" );
document.write( "\r\n" );
document.write( "Then we indicate the left multiplication of both sides by\r\n" );
document.write( "\"A%5E%28-1%29\" to get the equation \"A%5E%28-1%29%28A%2AX%29=A%5E%28-1%29B\":\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Next we use the associative principle to move the parentheses so that\r\n" );
document.write( "they are around the first two factors to get the equation \"%28A%5E%28-1%29%2AA%29%2AX=A%5E%28-1%29%2AB\":\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now we perform the actual multiplications and we get the equation \"IX=A%5E%28-1%29%2AB\":\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then when we multiply the identity matrix \"I\" by the column matrix of\r\n" );
document.write( "variables, we just get the matrix of variables, or the \r\n" );
document.write( "equation \"X=A%5E%28-1%29B\"\r\n" );
document.write( "\r\n" );
document.write( "\"%28matrix%283%2C1%2Cx%2Cy%2Cz%29%29=%28matrix%283%2C1%2C-1%2C4%2C-7%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "So \"x=-1\", \"y=4\", and \"z=-7\"\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );