document.write( "Question 1028468: Solve the linear system by row-reducing the corresponding augmented matrix:
\n" ); document.write( "-w+x-z=14
\n" ); document.write( "-w+x-y=-1
\n" ); document.write( "w-2x+y-z=4
\n" ); document.write( "-2w+x-y+4z=-19
\n" ); document.write( "b) Find the determinant of the coefficient matrix A.
\n" ); document.write( "c) Find abs( expr-(2/3)A^-1 A^t A ) \n" ); document.write( "
Algebra.Com's Answer #643663 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "\"system%28-w%2Bx-z=14%2C%0D%0A-w%2Bx-y=-1%2C%0D%0Aw-2x%2By-z=4%2C%0D%0A-2w%2Bx-y%2B4x=-19%29\"\r\n" );
document.write( "\r\n" );
document.write( "Write it this way:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "R1<->R3\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "R1+R2<->R2\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "R1+R3<->R3\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "2R1+R4->R4\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "-1R2->R2\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "R2+R3->R3\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "3R2+R4->R4\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "-1R3->R3\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "2R3+R4->R4\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Convert back to a system of equations:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "or\r\n" );
document.write( "\r\n" );
document.write( "\"system%28w-2x%2By-z=4%2C%0D%0Ax-y%2B2z=-18%2C%0D%0Ax-z=15%2C%0D%0A6z=-35%29\"\r\n" );
document.write( "\r\n" );
document.write( "Solve the bottom equation, get z=-35/6\r\n" );
document.write( "Substitute in 3rd equation, get x=17/6\r\n" );
document.write( "Substitute those in 2nd equation, get y=55/6\r\n" );
document.write( "Substitute those in 1st equation, get w=-16/3\r\n" );
document.write( "\r\n" );
document.write( "Coefficient matrix:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Determinant of coefficient matrix:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "R3+R2->R2\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "R4+R3->R3\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We expand about the 3rd column,\r\n" );
document.write( "which contains only -1 other than 0's\r\n" );
document.write( "\r\n" );
document.write( "R4+R3->R3\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We determine by the sign scheme whether\r\n" );
document.write( "we are to use the -1 as it is as a multiplier\r\n" );
document.write( "or whether we are to change its sign.  Here\r\n" );
document.write( "is the \"checkerboard\" sign scheme:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We find that that element -1 is in a \"-\" position\r\n" );
document.write( "in the checkerboard sign scheme, so we change the\r\n" );
document.write( "sign of the -1 to +1 and multiply it by the 3x3\r\n" );
document.write( "minor determinant gotten by omitting the row and \r\n" );
document.write( "column of the -1\r\n" );
document.write( "\r\n" );
document.write( "So changing the sign of -1 to 1 and multiplying it\r\n" );
document.write( "by the minor 3x3 determinant we have:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\"abs%28matrix%283%2C3%2C%0D%0A-1%2C1%2C-1%2C%0D%0A0%2C-1%2C-1%2C%0D%0A-1%2C-1%2C3%29%29\"\r\n" );
document.write( " \r\n" );
document.write( "-1*R1+R3->R3\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We expand about the 1st column, which contains \r\n" );
document.write( "only -1 other than 0's\r\n" );
document.write( "\r\n" );
document.write( "We determine by the sign scheme whether\r\n" );
document.write( "we are to use the -1 as it is as a multiplier\r\n" );
document.write( "or whether we are to change its sign.  Here\r\n" );
document.write( "is the \"checkerboard\" sign scheme:\r\n" );
document.write( "\"\"-\"\"\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We find that that element -1 is in a \"+\" position\r\n" );
document.write( "in the checkerboard sign scheme, so we leave the\r\n" );
document.write( "sign of -1 and multiply it by the 2x2\r\n" );
document.write( "minor determinant gotten by omitting the row and \r\n" );
document.write( "column of the -1\r\n" );
document.write( "\r\n" );
document.write( "multiplying the -1 by the minor 3x3 determinant \r\n" );
document.write( "we have:\r\n" );
document.write( "\r\n" );
document.write( "\"-1%2Aabs%28matrix%282%2C2%2C%0D%0A%0D%0A-1%2C-1%2C%0D%0A-2%2C4%29%29\"\"%22%22=%22%22\"\"-1%2A%28%28-1%29%284%29-%28-1%29%28-2%29%29\"\"%22%22=%22%22\"\"-1%28-4-2%29\"\"%22%22=%22%22\"(-1(-6)}}}\"%22%22=%22%22\"\"6\"\r\n" );
document.write( "\r\n" );
document.write( "I'll let you do the last one yourself. Here are\r\n" );
document.write( "A-inverse and A-transpose:\r\n" );
document.write( "\r\n" );
document.write( "  \r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );