document.write( "Question 174415: 
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document.write( "what is the value of this determinant?\r\n" );
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document.write( "                             thank you!
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Algebra.Com's Answer #129390 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "what is the value of this determinant?\r\n" );
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document.write( "                             thank you!\r\n" );
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document.write( "The third column has the most 0's, so\r\n" );
document.write( "we'll expand about it after we make it\r\n" );
document.write( "have all 0's but one.\r\n" );
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document.write( "Let's make the 3 in the 3rd column a 0\r\n" );
document.write( "by adding -3 times the 1st row to 1\r\n" );
document.write( "times the 4th row:\r\n" );
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document.write( "\"matrix%286%2C1%2C-3%2C%22%22%2C%22%22%2C1%2C%22%22%2C%22%22%29\"\r\n" );
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document.write( "Now we expand this 6x6 matrix by\r\n" );
document.write( "the 3rd column.  Since there is only\r\n" );
document.write( "1 non-zero element in the 3rd column,\r\n" );
document.write( "the 1 at the top.  We notice that it\r\n" );
document.write( "is in row 1 and column 3, so we add\r\n" );
document.write( "1+3 and get 4, and \"%28-1%29%5E4\"=+1 we\r\n" );
document.write( "multiply the 1 in the top row by +1\r\n" );
document.write( "and so the determinant is +1 times the\r\n" );
document.write( "5x5 determinant formed by removing\r\n" );
document.write( "both row 1 and column 3:\r\n" );
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document.write( "The first row has the most 0's, so\r\n" );
document.write( "we'll expand about it after we make it\r\n" );
document.write( "have all 0's but one.\r\n" );
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document.write( "Let's make the -1 in the 1st row 3rd \r\n" );
document.write( "column a 0 by adding 1 times the 2nd \r\n" );
document.write( "column to 1 times the 3rd column:\r\n" );
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document.write( "     1   1\r\n" );
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document.write( "Let's make the -1 in the 1st row 5th \r\n" );
document.write( "column a 0 by adding 1 times the 2nd \r\n" );
document.write( "column to 1 times the 5th column:\r\n" );
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document.write( "     1          1 \r\n" );
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document.write( "Now we expand this 5x5 matrix by\r\n" );
document.write( "the 1st row.  Since there is only\r\n" );
document.write( "1 non-zero element in the 1st row,\r\n" );
document.write( "the 1 at the 1st row 2nd column.\r\n" );
document.write( "Since it is in row 1 and column 2, so we add\r\n" );
document.write( "1+2 and get 3, and \"%28-1%29%5E3\"=-1 we\r\n" );
document.write( "multiply the 1 in the top row by -1\r\n" );
document.write( "and so the determinant is -1 times the\r\n" );
document.write( "4x4 determinant formed by removing\r\n" );
document.write( "both row 1 and column 2:\r\n" );
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document.write( "We can simplify by multiplying the -1\r\n" );
document.write( "by the 2nd row, changing their signs\r\n" );
document.write( "to positive:\r\n" );
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document.write( "Unfortunately there are no 0's at\r\n" );
document.write( "all.  So we'll make some.  The\r\n" );
document.write( "simplest row or column is the 1st\r\n" );
document.write( "column.  To make the 2 in the\r\n" );
document.write( "1st column a 0, add -2 times the\r\n" );
document.write( "1st row to 1 times the 2nd row:\r\n" );
document.write( "\"matrix%284%2C1%2C-2%2C1%2C%22%22%2C%22%22%29\"\r\n" );
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document.write( "Now since that 0 popped up\r\n" );
document.write( "over there in the second row,\r\n" );
document.write( "that makes us change plans, for\r\n" );
document.write( "now we have 2 0's in the 2nd row.\r\n" );
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document.write( "So we change plans and now we \r\n" );
document.write( "want to expand about the 2nd\r\n" );
document.write( "row.  So we can get a 0 where\r\n" );
document.write( "the -6 is by multiplying the\r\n" );
document.write( "3rd column by -6 and adding it\r\n" );
document.write( "to 1 times column 2.\r\n" );
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document.write( "Now we expand this 4x4 matrix by\r\n" );
document.write( "the 2nd row.  Since there is only\r\n" );
document.write( "1 non-zero element in the 1st row,\r\n" );
document.write( "the -1 at the 2nd row 3rd column.\r\n" );
document.write( "Since it is in row 2 and column 3, we add\r\n" );
document.write( "2+3 and get 5, and \"%28-1%29%5E5\"=-1, we\r\n" );
document.write( "multiply the -1 in the 2nd row 3rd column\r\n" );
document.write( "by -1 and so the determinant is 1 times the\r\n" );
document.write( "3x3 determinant formed by removing\r\n" );
document.write( "both row 2 and column 3:\r\n" );
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document.write( "The\r\n" );
document.write( "simplest row or column is the 2nd\r\n" );
document.write( "row.  To make the 7 a 0, we add\r\n" );
document.write( "-7 times the 1st column to 1 times \r\n" );
document.write( "the 3rd column:\r\n" );
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document.write( "-7   1\r\n" );
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document.write( "Now we expand this 3x3 matrix by\r\n" );
document.write( "the 2nd row.  Since there is only\r\n" );
document.write( "1 non-zero element in the 2nd row,\r\n" );
document.write( "the 1 at the 2nd row 1st column.\r\n" );
document.write( "Since it is in row 2 and column 1, we add\r\n" );
document.write( "2+1 and get 3, and \"%28-1%29%5E3\"=-1, we\r\n" );
document.write( "multiply the 1 in the 2nd row 1st column\r\n" );
document.write( "by -1 and so the determinant is -1 times the\r\n" );
document.write( "2x2 determinant formed by removing\r\n" );
document.write( "both row 2 and column 1:\r\n" );
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document.write( "\"-1abs%28matrix%282%2C2%2C%0D%0A+++-7%2C++-5%2C%0D%0A++-16%2C++-3%29%29\"\r\n" );
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document.write( "We can simplify by multiplying the -1\r\n" );
document.write( "by the 1st row, which changing the signs:\r\n" );
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document.write( "\"abs%28matrix%282%2C2%2C%0D%0A+++7%2C++5%2C%0D%0A++-16%2C++-3%29%29\"\r\n" );
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document.write( "To expand, it's just the difference of\r\n" );
document.write( "the product of the upper left to lower\r\n" );
document.write( "right diagonal and the product of the\r\n" );
document.write( "upper right to lower left diagonal.\r\n" );
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document.write( "(7)(-3)-(5)(-16)=-21+80=59\r\n" );
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document.write( "So the value of the determinant is 59.\r\n" );
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document.write( "Edwin

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