document.write( "Question 1105210: A= $\"a+1+1+1+1\"$
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\n" ); document.write( "\n" ); document.write( "(1)calculate det A ?
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Algebra.Com's Answer #720163 by ikleyn(52879) $\"\"$ $\"About$

You can put this solution on YOUR website!
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document.write( "A = \r\n" );
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document.write( "I will calculate the determinant det (A) in two steps.\r\n" );
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document.write( "Step 1\r\n" );
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document.write( "Subtract row 1 from the rows 2, 3, 4 and 5. You will get modified 5x5-matrix B\r\n" );
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document.write( "B = \r\n" );
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document.write( "Under these transformations (and after these transformations), det (A) = det (B).\r\n" );
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document.write( "Step 2\r\n" );
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document.write( "In matrix B, add all the columns 2, 3, 4 and 5 to the column 1. You will get modified 5x5-matrix C\r\n" );
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document.write( "C = \r\n" );
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document.write( "Under these transformations (and after these transformations), det (A) = det (B) = det (C).\r\n" );
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document.write( "Now calculate the determinant det (C) using cofactoring along the 1-st column.\r\n" );
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document.write( "You will get  det (C) = .\r\n" );
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document.write( "Thus  det (A) = .\r\n" );
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document.write( "In this way, you can calculate the determinant of the given form matrix for ANY order \"n\".\r\n" );
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document.write( "So, the question #1 is answered.\r\n" );
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document.write( "The answer to the question #2 is:  \r\n" );
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document.write( "    Ax = 0 has nonzero solutions for a = -4  and for a = 1.\r\n" );
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document.write( "The question #3 is formulated INCORRECTLY, in my view.\r\n" );
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document.write( "The correct formulation is THIS:\r\n" );
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document.write( "    when Ax = 0 has nonzero solution, how many arbitrary constants are there in the general solution?\r\n" );
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document.write( "And the answer to this question is:\r\n" );
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document.write( "    when a = 1,  the general solution has  4  arbitrary constants;\r\n" );
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document.write( "    when a = -4, the general solution has  1 arbitrary constant.\r\n" );
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