document.write( "Question 1121869: Find the determinant of the following matrices using co-factor method.\r
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Algebra.Com's Answer #737870 by ikleyn(52810) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Answer. The determinant is equal to 0 (zero, ZERO).\r
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document.write( "It is WRONG (=very ineffective) way to calculate this determinant using co-factor method.\r\n" );
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document.write( "You can get the answer MUCH FASTER and with minimum calculations (actually, without massive calculations) using properties of determinant.\r\n" );
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document.write( "a)  Replace the second row of the matrix by the sum of the first and the second rows of the given matrix.\r\n" );
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document.write( "    Then the second row becomes  (5, 5, 5, 5).\r\n" );
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document.write( "    Determinant of the transformed matrix will be the same as for the original matrix, by the fundamental property of the determinant.\r\n" );
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document.write( "b)  Next, replace the fourth row of the matrix by the sum of the third and the fourth rows of the given matrix.\r\n" );
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document.write( "    Then the fourth row becomes  (5, 5, 5, 5).\r\n" );
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document.write( "    Determinant of the transformed matrix will be the same as for the original matrix, by the fundamental property of the determinant.\r\n" );
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document.write( "c)  Now the transformed matrix contains two identical rows, the second and the fourth, consisting of four \"5\",  (5,5,5,5).\r\n" );
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document.write( "    Hence, due to the other fundamental property, the determinant of the transformed matrix is zero.\r\n" );
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document.write( "    Therefore, the determinant of the original matrix is equal to 0 (zero, ZERO).\r\n" );
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\n" ); document.write( "\n" ); document.write( "On these fundamental properties of determinant see the site\r
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\n" ); document.write( "\n" ); document.write( " - https://www.math.drexel.edu/~jwd25/LM_SPRING_07/lectures/lecture4B.html\r
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\n" ); document.write( "\n" ); document.write( "https://www.math.drexel.edu/~jwd25/LM_SPRING_07/lectures/lecture4B.html\r
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\n" ); document.write( "\n" ); document.write( "You can find these properties in many sites in the Internet, as well as in your textbook (textbooks) in High/Abstract/Linear Algebra.\r
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