document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=211750>Question 211750</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi all, I am struggling a bit with the following matrices problem.<BR>\n" );
document.write( "I need to find the determinate of the following matrices by Inspection. Then give reasons in each each. How do I do this?<BR>\n" );
document.write( "The matrices are:<BR>\n" );
document.write( "B = <BR>\n" );
document.write( "( 2 11 -3 1)<BR>\n" );
document.write( "( 1 5  -7 -4)<BR>\n" );
document.write( "( 6 13 -5 2)<BR>\n" );
document.write( "( 4 22 -6 2)<BR>\n" );
document.write( "C =<BR>\n" );
document.write( "( 2 9 -1)<BR>\n" );
document.write( "( 0 2  7)<BR>\n" );
document.write( "( 0 0  4)<BR>\n" );
document.write( "D =<BR>\n" );
document.write( "( -3 0 0)<BR>\n" );
document.write( "( 0 2 0)<BR>\n" );
document.write( "( 0 0 -5)\r<BR>\n" );
document.write( "\n" );
document.write( "Any help would be great.<BR>\n" );
document.write( "-Nick.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #159986 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=159986\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> For matrix B, the key is to notice that row 4 is exactly twice row 1. In such a matrix the determinant is always zero.<br><BR>\n" );
document.write( "Both matrices C and D are diagonal matrices (i.e. a matrix with nothing but zeros above and/or below the main diagonal). In diagonal matrices the determinant is simply the product of the numbers in the diagonal. For C this would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A2%2A4+=+16 ALIGN=MIDDLE  ALT=\"2%2A2%2A4+=+16\"   DISABLED_style= \"max-width:90%; height:auto;\" >. For D this would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%2A2%2A-5+=+30 ALIGN=MIDDLE  ALT=\"-3%2A2%2A-5+=+30\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "If you need more \"reasons\"...<br><BR>\n" );
document.write( "For matrix B, think about adding -2 times the first row to the fourth row. This type of operation does not change the value of the determinant and, in this casse, it results in row 4 becoming all zeros. Then think about finding the value of the determinant using the fourth row. Result: 0.<br><BR>\n" );
document.write( "For matrices C and D, think about using the first column to find the value of the determinant. Since there is only one non-zero value in this column, it should be easy to visualize the calculations and see why only the product of the numbers in the diagonal will matter. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );