document.write( "Question 204044: Compute the determinant
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document.write( "| 1  2  3  4|\r\n" );
document.write( "| 5  6  7  8|\r\n" );
document.write( "| 9 10 11 12|\r\n" );
document.write( "|13 14 15 16|
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\n" ); document.write( "\n" ); document.write( "I need help with four equations. I can do it with three. Thank you. \n" ); document.write( "
Algebra.Com's Answer #153980 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "| 1  2  3  4|\r\n" );
document.write( "| 5  6  7  8|\r\n" );
document.write( "| 9 10 11 12|\r\n" );
document.write( "|13 14 15 16|\r\n" );
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document.write( "Get a 0 where the 2 is.  To do that, \r\n" );
document.write( "multiplying the first column through \r\n" );
document.write( "by -2:\r\n" );
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document.write( " -2 \r\n" );
document.write( "-10 \r\n" );
document.write( "-18\r\n" );
document.write( "-26\r\n" );
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document.write( "then add that to the second column:\r\n" );
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document.write( " -2 +  2 =   0 \r\n" );
document.write( "-10 +  6 =  -4\r\n" );
document.write( "-18 + 10 =  -8 \r\n" );
document.write( "-26 + 14 = -12\r\n" );
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document.write( "Replace the 2nd column by that:\r\n" );
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document.write( "| 1   0  3  4|\r\n" );
document.write( "| 5  -4  7  8|\r\n" );
document.write( "| 9  -8 11 12|\r\n" );
document.write( "|13 -12 15 16|\r\n" );
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document.write( "Get a 0 where the 3 is.  To do that, \r\n" );
document.write( "multiplying the first column through \r\n" );
document.write( "by -3:\r\n" );
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document.write( " -3 \r\n" );
document.write( "-15 \r\n" );
document.write( "-27\r\n" );
document.write( "-39\r\n" );
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document.write( "then add that to the third column:\r\n" );
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document.write( " -3 +  3 =   0 \r\n" );
document.write( "-15 +  7 =  -8\r\n" );
document.write( "-27 + 11 = -16 \r\n" );
document.write( "-39 + 15 = -24\r\n" );
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document.write( "Replace the 3rd column by that:\r\n" );
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document.write( "| 1   0   0  4|\r\n" );
document.write( "| 5  -4  -8  8|\r\n" );
document.write( "| 9  -8 -16 12|\r\n" );
document.write( "|13 -12 -24 16|\r\n" );
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document.write( "Get a 0 where the 4 is.  To do that, \r\n" );
document.write( "multiplying the first column through \r\n" );
document.write( "by -4:\r\n" );
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document.write( " -4 \r\n" );
document.write( "-20 \r\n" );
document.write( "-36\r\n" );
document.write( "-52\r\n" );
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document.write( "then add that to the fourth column:\r\n" );
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document.write( " -4 +  4 =   0 \r\n" );
document.write( "-10 +  8 =  -2\r\n" );
document.write( "-18 + 12 =  -6 \r\n" );
document.write( "-26 + 16 = -10\r\n" );
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document.write( "Replace the 4th column by that:\r\n" );
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document.write( "| 1   0   0   0|\r\n" );
document.write( "| 5  -4  -8  -2|\r\n" );
document.write( "| 9  -8 -16  -6|\r\n" );
document.write( "|13 -12 -24 -10|\r\n" );
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document.write( "Now expand about the top row.\r\n" );
document.write( "Since there are 3 0's in the\r\n" );
document.write( "top row, we only need multiply\r\n" );
document.write( "the 1 by its signed minor.\r\n" );
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document.write( "The 1 is in row 1 and column 1,\r\n" );
document.write( "adding the row and column numbers\r\n" );
document.write( "gives 2, which is even, so the\r\n" );
document.write( "sign of the minor is +. The \r\n" );
document.write( "minor of 1 is the 3x3 determinant\r\n" );
document.write( "formed by taking all the elements\r\n" );
document.write( "which are NOT in the same row or\r\n" );
document.write( "column as the 1, so the above becomes\r\n" );
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document.write( "    | -4  -8  -2|\r\n" );
document.write( "1 × | -8 -16  -6|\r\n" );
document.write( "    |-12 -24 -10|\r\n" );
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document.write( "Now you can evaluate that 3×3 determinant,\r\n" );
document.write( "and you'll find it is 0.  So the value of the\r\n" );
document.write( "original 4×4 matrix is 0.\r\n" );
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document.write( "Edwin
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