document.write( "Question 1167475: How does the rank of the following matrix depend on the value of t?
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Algebra.Com's Answer #792062 by ikleyn(52781) $\"\"$ $\"About$

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document.write( "Doing formally, you need calculate the determinant of the matrix as a polynomial of \"t\";\r\n" );
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document.write( "                    then find its roots (zeroes);\r\n" );
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document.write( "                    then analyze the matrix at every root value of t.\r\n" );
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document.write( "But, looking in this matrix, I see it MENTALLY, that one of the root is 1 (one, ONE)\r\n" );
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document.write( "    since at t= 1 all three lines of the matrix are identical, making the determinant equal to zero.\r\n" );
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document.write( "Also, due to the same reason, at t= 1  the rank of the matrix is 1.\r\n" );
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document.write( "It means, from the other side, that  t= 1  is the root of the determinant of the multiplicity 2.\r\n" );
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document.write( "Next, if you add the first and the second rows of the matrix to its third row,\r\n" );
document.write( "you will get in the third row three\r\n" );
document.write( "identical elements (t+2), (t+2), (t+2)\r\n" );
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document.write( "It means that t= -2 is the third root of the determinant of the multiplicity 1.\r\n" );
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document.write( "ANSWER.  The determinant is zero at t= 1  and  t= -2.\r\n" );
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document.write( "         The value of t= 1 is the zero of multiplicity 2;  the value of t= -2 is the zero of the multiplicity 1.\r\n" );
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document.write( "         The rank of the matrix is 1 at t= 1;  is 2 at t= -2  and is 3 at any other value of t.\r\n" );
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