document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200083>Question 1200083</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let A be the matrix of coefficients of a 5 × 7 system of linear equations, A⃗x = ⃗b. Using row operations, you find that A is row equivalent to a matrix in reduced row echelon form with one row of zeroes at the bottom.<BR>\n" );
document.write( "(a)  What is rank(A)?<BR>\n" );
document.write( "(b)  How many free variables does the system have?<BR>\n" );
document.write( "(c)  For the given system how many possible solutions could it have? (Circle all which apply)<BR>\n" );
document.write( "1. 0 solutions<BR>\n" );
document.write( "2. 1 solution<BR>\n" );
document.write( "3. infinite solutions<BR>\n" );
document.write( "(d) For the associated homogeneous system A⃗x = ⃗0, how many possible solutions could it have? (Circle all which apply)<BR>\n" );
document.write( "1. 0 solutions<BR>\n" );
document.write( "2. 1 solution<BR>\n" );
document.write( "3. infinite solutions </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848091 by </B><A HREF=/tutors/aboutme.mpl?userid=GingerAle><B>GingerAle(43)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=GingerAle><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848091\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Rank(A)**\r<BR>\n" );
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document.write( "* The rank of a matrix is the number of non-zero rows in its row-echelon form. <BR>\n" );
document.write( "* Since A has one row of zeros and is row equivalent to a matrix in reduced row echelon form, its rank is 4.\r<BR>\n" );
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document.write( "**b) Number of Free Variables**\r<BR>\n" );
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document.write( "* The number of free variables is equal to the number of columns minus the rank of the matrix.<BR>\n" );
document.write( "* Number of free variables = 7 (columns) - 4 (rank) = 3\r<BR>\n" );
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document.write( "**c) Possible Solutions for A⃗x = ⃗b**\r<BR>\n" );
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document.write( "* **Possible Solutions:**<BR>\n" );
document.write( "    * **Infinite solutions** <BR>\n" );
document.write( "    * **No solutions**\r<BR>\n" );
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document.write( "* **Explanation:**<BR>\n" );
document.write( "    * If the last row in the reduced row echelon form of the augmented matrix [A | ⃗b] is of the form [0 0 0 | c] where c is a non-zero constant, then the system has no solution. <BR>\n" );
document.write( "    * Otherwise, if the last row is all zeros, the system will have infinite solutions due to the free variables.\r<BR>\n" );
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document.write( "**d) Possible Solutions for A⃗x = ⃗0**\r<BR>\n" );
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document.write( "* **Possible Solutions:**<BR>\n" );
document.write( "    * **Infinite solutions**\r<BR>\n" );
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document.write( "* **Explanation:**<BR>\n" );
document.write( "    * For the homogeneous system A⃗x = ⃗0, the last row in the augmented matrix [A | ⃗0] will always be all zeros. <BR>\n" );
document.write( "    * Since there are free variables, the homogeneous system will always have infinite solutions (including the trivial solution ⃗x = ⃗0).\r<BR>\n" );
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document.write( "**In Summary:**\r<BR>\n" );
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document.write( "* **Rank(A) = 4**<BR>\n" );
document.write( "* **Number of Free Variables = 3**<BR>\n" );
document.write( "* **Possible Solutions for A⃗x = ⃗b:** Infinite solutions or no solutions<BR>\n" );
document.write( "* **Possible Solutions for A⃗x = ⃗0:** Infinite solutions<BR>\n" );
document.write( " </SPAN>\n" );
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