document.write( "Question 1055331: Find the value(s)of b for which Z = \r
\n" ); document.write( "\n" ); document.write( "6 −b −4
\n" ); document.write( "b 0 1
\n" ); document.write( "1 −2 1\r
\n" ); document.write( "\n" ); document.write( " is singular\r
\n" ); document.write( "\n" ); document.write( "i worked it but not sure of the answer,
\n" ); document.write( "please show me the steps to get the answer.
\n" ); document.write( "thank you \n" ); document.write( "
Algebra.Com's Answer #670580 by ikleyn(53765)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "Find the value(s)of b for which Z = \r
\n" ); document.write( "\n" ); document.write( "6 −b −4
\n" ); document.write( "b 0 1
\n" ); document.write( "1 −2 1\r
\n" ); document.write( "\n" ); document.write( " is singular\r
\n" ); document.write( "\n" ); document.write( "i worked it but not sure of the answer,
\n" ); document.write( "please show me the steps to get the answer.
\n" ); document.write( "thank you
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r
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document.write( "Z = \"%28matrix%283%2C3%2C+6%2C-b%2C-4%2C++b%2C0%2C1%2C++1%2C-2%2C1%29%29\".\r\n" );
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document.write( "det(Z) = [6*0*1 + (-b)*1*1 + b*(-2)*(-4)] - [1*0*(-4) + b*(-b)*1 + 6*(-2)*1] = \r\n" );
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document.write( "          0 - b + 8b + 0 + b^2 + 12 = b^2 + 7b + 12.\r\n" );
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document.write( "The matrix is singular if and only if det(Z) = 0, i.e.\r\n" );
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document.write( "b^2 + 7b + 12 = 0.\r\n" );
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document.write( "Factor the polynomial in the left side:\r\n" );
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document.write( "(b-3)*(b-4) = 0.\r\n" );
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document.write( "The roots are b = 3  and  b = 4.\r\n" );
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document.write( "The matrix Z is singular if and only if b = 3  or  b = 4.\r\n" );
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\n" ); document.write( "\n" ); document.write( "On calculating determinants of 3x3 matrices see the lesson\r
\n" ); document.write( "\n" ); document.write( "    - Determinant of a 3x3 matrix \r
\n" ); document.write( "\n" ); document.write( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic
\n" ); document.write( "     \"3x3-Matrices, determinants, Cramer's rule for systems in three unknowns\" \r
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