document.write( "Question 92867This question is from textbook Finite Mathematics
\n" ); document.write( ": The inverse of 1 -1 0
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Algebra.Com's Answer #67683 by Edwin McCravy(20060) $\"\"$ $\"About$
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document.write( "Find the inverse of\r\n" );
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document.write( "[1 -1  0]\r\n" );
document.write( "[0  1  1]\r\n" );
document.write( "[1  0 -1]\r\n" );
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document.write( "Put the identity matrix on the right\r\n" );
document.write( "(The identity matrix is a 3x3 matrix\r\n" );
document.write( "which has 1's on the main diagonal and\r\n" );
document.write( "0's elsewhere):\r\n" );
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document.write( "[1 -1  0 | 1  0  0]\r\n" );
document.write( "[0  1  1 | 0  1  0]\r\n" );
document.write( "[1  0 -1 | 0  0  1]\r\n" );
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document.write( "Make the identity matrix on the left.\r\n" );
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document.write( "Get a 0 where the first 1 is in the bottom row:\r\n" );
document.write( "multiply the 1st row by -1, add it to 1 times\r\n" );
document.write( "the 3rd row and restore the 1st row:\r\n" );
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document.write( "-1[1 -1  0 | 1  0  0]\r\n" );
document.write( "  [0  1  1 | 0  1  0]\r\n" );
document.write( " 1[1  0 -1 | 0  0  1]\r\n" );
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document.write( "  [1 -1  0 | 1  0  0]\r\n" );
document.write( "  [0  1  1 | 0  1  0]\r\n" );
document.write( "  [0  1 -1 |-1  0  1]\r\n" );
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document.write( "Get a 0 where the -1 is in the top row:\r\n" );
document.write( "multiply the 2nd row by 1, add it to 1 times\r\n" );
document.write( "the 1st row and restore the 2nd row: \r\n" );
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document.write( " 1[1 -1  0 | 1  0  0]\r\n" );
document.write( " 1[0  1  1 | 0  1  0]\r\n" );
document.write( "  [0  1 -1 |-1  0  1]\r\n" );
document.write( "\r\n" );
document.write( "  [1  0  1 | 1  1  0]\r\n" );
document.write( "  [0  1  1 | 0  1  0]\r\n" );
document.write( "  [0  1 -1 |-1  0  1]\r\n" );
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document.write( "Get a 0 where the first 1 is in the bottom row:\r\n" );
document.write( "multiply the 2nd row by -1, add it to 1 times\r\n" );
document.write( "the 3rd row and restore the 2nd row:\r\n" );
document.write( "\r\n" );
document.write( "  [1  0  1 | 1  1  0]\r\n" );
document.write( "-1[0  1  1 | 0  1  0]\r\n" );
document.write( " 1[0  1 -1 |-1  0  1]\r\n" );
document.write( "\r\n" );
document.write( "  [1  0  1 | 1  1  0]\r\n" );
document.write( "  [0  1  1 | 0  1  0]\r\n" );
document.write( "  [0  0 -2 |-1 -1  1]\r\n" );
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document.write( "Get a 0 where the second 1 is in the 1st row:\r\n" );
document.write( "multiply the 3rd row by 1, add it to 2 times\r\n" );
document.write( "the 1st row and restore the 3rd row:\r\n" );
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document.write( "2[1  0  1 | 1  1  0]\r\n" );
document.write( " [0  1  1 | 0  1  0]\r\n" );
document.write( "1[0  0 -2 |-1 -1  1]\r\n" );
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document.write( " [2  0  0 | 1  1  1]\r\n" );
document.write( " [0  1  1 | 0  1  0]\r\n" );
document.write( " [0  0 -2 |-1 -1  1]\r\n" );
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document.write( "Get a 0 where the second 1 is in the 2nd row:\r\n" );
document.write( "multiply the 3rd row by 1, add it to 2 times\r\n" );
document.write( "the 2nd row and restore the 3rd row:\r\n" );
document.write( "\r\n" );
document.write( " [2  0  0 | 1  1  1]\r\n" );
document.write( "2[0  1  1 | 0  1  0]\r\n" );
document.write( "1[0  0 -2 |-1 -1  1]\r\n" );
document.write( "\r\n" );
document.write( " [2  0  0 | 1  1  1]\r\n" );
document.write( " [0  2  0 |-1  1  1]\r\n" );
document.write( " [0  0 -2 |-1 -1  1]\r\n" );
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document.write( "To get a 1 where the 2 is in the 1st row,\r\n" );
document.write( "multiply the 1st row by .5\r\n" );
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document.write( ".5[2  0  0 | 1  1  1]\r\n" );
document.write( "  [0  2  0 |-1  1  1]\r\n" );
document.write( "  [0  0 -2 |-1 -1  1]\r\n" );
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document.write( "  [1  0  0 |.5 .5 .5]\r\n" );
document.write( "  [0  2  0 |-1  1  1]\r\n" );
document.write( "  [0  0 -2 |-1 -1  1]\r\n" );
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document.write( "To get a 1 where the 2 is in the 2nd row,\r\n" );
document.write( "multiply the 2nd row by .5\r\n" );
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document.write( "  [1  0  0 |.5 .5 .5]\r\n" );
document.write( ".5[0  2  0 |-1  1  1]\r\n" );
document.write( "  [0  0 -2 |-1 -1  1]\r\n" );
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document.write( "  [1  0  0 | .5 .5 .5]\r\n" );
document.write( "  [0  1  0 |-.5 .5 .5]\r\n" );
document.write( "  [0  0 -2 | -1 -1  1]\r\n" );
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document.write( "To get a 1 where the -2 is in the 3rd row,\r\n" );
document.write( "multiply the 2nd row by -.5\r\n" );
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document.write( "  1[1  0  0 | .5 .5 .5]\r\n" );
document.write( "   [0  1  0 |-.5 .5 .5]\r\n" );
document.write( "-.5[0  0 -2 | -1 -1  1]\r\n" );
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document.write( "  [1  0  0 | .5 .5  .5]\r\n" );
document.write( "  [0  1  0 |-.5 .5  .5]\r\n" );
document.write( "  [0  0  1 | .5 .5 -.5]\r\n" );
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document.write( "Now that we have the identity on the left,\r\n" );
document.write( "the matrix on the right is the inverse:\r\n" );
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document.write( "           [ .5 .5  .5]\r\n" );
document.write( "           [-.5 .5  .5]\r\n" );
document.write( "           [ .5 .5 -.5]\r\n" );
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document.write( "Edwin
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