document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=191470>Question 191470</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let A =\r<BR>\n" );
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document.write( "1 -2 -1<BR>\n" );
document.write( "0  1  2<BR>\n" );
document.write( "0  0 -1\r<BR>\n" );
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document.write( "A. Calculate the determinant of A.<BR>\n" );
document.write( "Does the inverse matrix exist? If so, then calculate Ainverse. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #143674 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=143674\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Since we have a upper triangular matrix, this means that the determinant is simply the product of all of the diagonal entries. So \r<BR>\n" );
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document.write( "det(A)=(1)(1)(-1)=-1\r<BR>\n" );
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document.write( "So the determinant of A is -1\r<BR>\n" );
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document.write( "Since the determinant is NOT equal to zero, this means that the inverse of A exists. To find the inverse of A, you have many options, but the best option (in my opinion) is to row reduce the augmented matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AI ALIGN=MIDDLE  ALT=\"AI\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So append the 3x3 matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C1%2C0%2C0%2C0%2C1%2C0%2C0%2C0%2C1%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C1%2C0%2C0%2C0%2C1%2C0%2C0%2C0%2C1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%283%2C3%2C1%2C-2%2C-1%2C0%2C1%2C2%2C0%2C0%2C-1%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%283%2C3%2C1%2C-2%2C-1%2C0%2C1%2C2%2C0%2C0%2C-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C6%2C1%2C-2%2C-1%2C1%2C0%2C0%2C0%2C1%2C2%2C0%2C1%2C0%2C0%2C0%2C-1%2C0%2C0%2C1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "From there, just row reduce the 3x6 matrix to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E%28-1%29 ALIGN=MIDDLE  ALT=\"A%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Let me know if you need help with the row reduction.\r<BR>\n" );
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document.write( "Note: you should get the answer: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E%28-1%29=%28matrix%283%2C3%2C1%2C2%2C3%2C0%2C1%2C2%2C0%2C0%2C-1%29%29 ALIGN=MIDDLE  ALT=\"A%5E%28-1%29=%28matrix%283%2C3%2C1%2C2%2C3%2C0%2C1%2C2%2C0%2C0%2C-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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