document.write( "Question 1062852: a square matrix A is said to be idempotent A^2=A
\n" ); document.write( "(a)give an example of idempotent matrix other than 0 and I
\n" ); document.write( "(b)show that,if matrix A is both idempotent and invertible then A=I \n" ); document.write( "
Algebra.Com's Answer #677866 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "Just the (a) part:\r\n" );
document.write( "\r\n" );
document.write( "Let A = \"%28matrix%282%2C2%2Cw%2Cx%2Cy%2Cz%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "be idempotent.  Then\r\n" );
document.write( "\r\n" );
document.write( "   \"A%5E2\"\"%22%22=%22%22\"\"A\"\r\n" );
document.write( "\r\n" );
document.write( " \"%28matrix%282%2C2%2Cw%2Cx%2Cy%2Cz%29%29%5E2\"\"%22%22=%22%22\"\"%28matrix%282%2C2%2Cw%2Cx%2Cy%2Cz%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"%28matrix%282%2C2%2Cw%2Cx%2Cy%2Cz%29%29%2A%28matrix%282%2C2%2Cw%2Cx%2Cy%2Cz%29%29\"\"%22%22=%22%22\"\"%28matrix%282%2C2%2Cw%2Cx%2Cy%2Cz%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"%28matrix%282%2C2%2Cw%5E2%2Bxy%2Cwx%2Bxz%2Cwy%2Byz%2Cxy%2Bz%5E2%29%29\"\"%22%22=%22%22\"\"%28matrix%282%2C2%2Cw%2Cx%2Cy%2Cz%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "So we set each elements on the left equal to the corresponding\r\n" );
document.write( "element on the right:\r\n" );
document.write( "\r\n" );
document.write( "\"system%28w%5E2%2Bxy=w%2C+wx%2Bxz=x%2C+wy%2Byz=y%2C+xy%2Bz%5E2=z%29\"\r\n" );
document.write( "\r\n" );
document.write( "Divide the second one through by x or the third one through\r\n" );
document.write( "by y:\r\n" );
document.write( "\r\n" );
document.write( "w + z = 1 \r\n" );
document.write( "\r\n" );
document.write( "So we pick, say w = 1/4 and z = 3/4 since they have sum 1 \r\n" );
document.write( "\r\n" );
document.write( "Substituting in\r\n" );
document.write( "\r\n" );
document.write( "\"w%5E2%2Bxy=w\"\r\n" );
document.write( "\r\n" );
document.write( "\"%281%2F4%29%5E2%2Bxy=1%2F4\"\r\n" );
document.write( "\r\n" );
document.write( "\"1%2F16%2Bxy=1%2F4\"\r\n" );
document.write( "\r\n" );
document.write( "\"1%2B16xy=4\"\r\n" );
document.write( " \"16xy=3\"\r\n" );
document.write( " \"xy=3%2F16\"\r\n" );
document.write( "\r\n" );
document.write( "Choose x=1 and y=3/16, because their product is 3/16.\r\n" );
document.write( "\r\n" );
document.write( "Substitute in\r\n" );
document.write( "\r\n" );
document.write( "A = \"%28matrix%282%2C2%2Cw%2Cx%2Cy%2Cz%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "A = \"%28matrix%282%2C4%2C1%2F4%2C%22%22%2C%22%22%2C1%2C3%2F16%2C%22%22%2C%22%22%2C3%2F4%29%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );