\r\n" );
document.write( "Augment the matrix with the identity matrix on the right.\r\n" );
document.write( "Then do row operations to get the identity on the left,\r\n" );
document.write( "and the inverse will be on the right:\r\n" );
document.write( "
\r\n" );
document.write( "-----------------------------------------------\r\n" );
document.write( "Swap rows 1 and 2 to get a non-zero element\r\n" );
document.write( "in the upper left corner:\r\n" );
document.write( "
\r\n" );
document.write( "-----------------------------------------------\r\n" );
document.write( "We get column 1 so that all its elements are\r\n" );
document.write( "0 except the element in row 1. \r\n" );
document.write( "\r\n" );
document.write( "The least common multiple of the non-zero \r\n" );
document.write( "elements in column 1 is 2, so we get all\r\n" );
document.write( "the non-zero elements in column 1 to be 2\r\n" );
document.write( "except row 1, and we get it to be -2\r\n" );
document.write( "So we only need to\r\n" );
document.write( "multiply row 1 by -2\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Add row 1 to rows 3 and 4, and restore\r\n" );
document.write( "row 1:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "-----------------------------------------------\r\n" );
document.write( "We get column 2 so that all its elements are\r\n" );
document.write( "0 except the element in row 2. \r\n" );
document.write( "\r\n" );
document.write( "The least common multiple of the non-zero \r\n" );
document.write( "elements in column 2 is 1, so we get all\r\n" );
document.write( "the non-zero elements in column 2 to be 1\r\n" );
document.write( "except row 2, and we get it to be -1\r\n" );
document.write( "So we only need to\r\n" );
document.write( "multiply row 2 by -1\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Add row 2 to rows 1 and 4, and restore\r\n" );
document.write( "row 2:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "-----------------------------------------------\r\n" );
document.write( "We get column 3 so that all its elements are\r\n" );
document.write( "0 except the element in row 3. \r\n" );
document.write( "\r\n" );
document.write( "The least common multiple of the non-zero \r\n" );
document.write( "elements in column 3 is 6, so we get all\r\n" );
document.write( "the non-zero elements in column 3 to be 6\r\n" );
document.write( "except row 3, and we get it to be -6\r\n" );
document.write( "So we need to\r\n" );
document.write( "multiply row 2 by 3\r\n" );
document.write( "multiply row 3 by 3\r\n" );
document.write( "multiply row 4 by -2\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Add row 3 to rows 2 and 4, and \r\n" );
document.write( "restore row 3 \r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "-----------------------------------------------\r\n" );
document.write( "We get column 4 so that all its elements are\r\n" );
document.write( "0 except the element in row 4. \r\n" );
document.write( "\r\n" );
document.write( "The least common multiple of the non-zero \r\n" );
document.write( "elements in column 4 is 3, so we get all\r\n" );
document.write( "the non-zero elements in column 4 to be 3\r\n" );
document.write( "except in row 4, and we get it to be -3\r\n" );
document.write( "So we need to\r\n" );
document.write( "multiply row 1 by 3\r\n" );
document.write( "multiply row 2 by -1\r\n" );
document.write( "multiply row 3 by -1\r\n" );
document.write( "multiply row 4 by -3\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Add row 4 to rows 1, 2, and 3, and restore row 4\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Get 1's on the diagonal\r\n" );
document.write( "\r\n" );
document.write( "Divide row 1 by 3\r\n" );
document.write( "Divide row 2 by -3\r\n" );
document.write( "Divide row 3 by 2\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "-------------------------------------\r\n" );
document.write( "Now that we have the identity matrix on the left\r\n" );
document.write( "of the petition, the inverse of the given matrix \r\n" );
document.write( "is the 4×4 matrix on the right of the petition:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" );
document.write( "![\"\"](\"/images/stars/stars-13.gif\")
