document.write( "Question 1161347: Let M=[4 4 ; −2 10].
\n" ); document.write( "Find formulas for the entries of M^n, where n is a positive integer. \n" ); document.write( "
Algebra.Com's Answer #784948 by Edwin McCravy(20065)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "We diagnalize the matrix:\r\n" );
document.write( "\r\n" );
document.write( "\"matrix%281%2C3%2CM%2C%22%22=%22%22%2C%0D%0A%28matrix%282%2C2%2C4%2C4%2C-2%2C10%29%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "We find the eigenvalues\r\n" );
document.write( "\r\n" );
document.write( "\"abs%28matrix%282%2C2%2C4-lambda%2C4%2C-2%2C10-lambda%29%29%29\"\"%22%22=%22%22\"\"0\"\r\n" );
document.write( "\r\n" );
document.write( "\"%284-lambda%29%2810-lambda%29-%28-2%29%284%29=0\"\r\n" );
document.write( "\r\n" );
document.write( "\"40-14lambda%2Blambda%5E2%2B8=0\"\r\n" );
document.write( "\r\n" );
document.write( "\"lambda%5E2-14lambda%2B48=0\"\r\n" );
document.write( "\r\n" );
document.write( "\"%28lambda-6%29%28lambda-8%29=0\"\r\n" );
document.write( "\r\n" );
document.write( "λ-6=0;  λ-8=0\r\n" );
document.write( "  λ=6;    λ=8\r\n" );
document.write( "\r\n" );
document.write( "by writing it as \r\n" );
document.write( "\r\n" );
document.write( "\"matrix%281%2C3%2CM%2C%22%22=%22%22%2CPDP%5E%28-1%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "where D is the diagonal matrix with the two eigenvalues on the \r\n" );
document.write( "main diagonal:\r\n" );
document.write( "\r\n" );
document.write( "\"D+=+%28matrix%282%2C2%2C6%2C0%2C0%2C8%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "and the matrix P is \r\n" );
document.write( "\r\n" );
document.write( "\"P=%28matrix%281%2C2%2CV%5B1%5D%2CV%5B2%5D%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "where the V's are the two column eigenvectors for the two eigenvalues\r\n" );
document.write( "\r\n" );
document.write( "We find V1 which is the eigengvector for the eigenvalue λ=6.\r\n" );
document.write( "\r\n" );
document.write( "We find solutions for\r\n" );
document.write( "\r\n" );
document.write( "\"%28M-6I%29X=0\"\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\"-2x%5B1%5D%2B4x%5B2%5D=0\"\r\n" );
document.write( "Divide thru by -2\r\n" );
document.write( "\"x%5B1%5D-2x%5B2%5D=0\"\r\n" );
document.write( "\"x%5B1%5D=2x%5B2%5D\"\r\n" );
document.write( "\r\n" );
document.write( "We can take x1=1 and x1=1\r\n" );
document.write( "\r\n" );
document.write( "So \r\n" );
document.write( "\r\n" );
document.write( "\"v%5B1%5D=%28matrix%282%2C1%2C2%2C1%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "Now we do the same for the other eigenvalue\r\n" );
document.write( "\r\n" );
document.write( "---\r\n" );
document.write( "\r\n" );
document.write( "We find solutions for\r\n" );
document.write( "\r\n" );
document.write( "\"%28M-8I%29X=0\"\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\"-4x%5B1%5D%2B4x%5B2%5D=0\"\r\n" );
document.write( "Divide thru by -4\r\n" );
document.write( "\"x%5B1%5D-x%5B2%5D=0\"\r\n" );
document.write( "\"x%5B1%5D=x%5B2%5D\"\r\n" );
document.write( "\r\n" );
document.write( "We can take x1=1 and x2=1\r\n" );
document.write( "\r\n" );
document.write( "So \r\n" );
document.write( "\r\n" );
document.write( "\"v%5B1%5D=%28matrix%282%2C1%2C1%2C1%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "So\r\n" );
document.write( "\r\n" );
document.write( "\"P=%28matrix%282%2C2%2C2%2C1%2C1%2C1%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "And since the determinant of P is 1, to find P-1 we only\r\n" );
document.write( "need to swap the elements on the the main diagonal and change the\r\n" );
document.write( "signs of the other two elements\"\r\n" );
document.write( "\r\n" );
document.write( "\"P%5E%28-1%29=%28matrix%282%2C2%2C1%2C-1%2C1%2C2%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "Then \r\n" );
document.write( "\r\n" );
document.write( "\"M=PDP%5E%28-1%29\"\r\n" );
document.write( "\r\n" );
document.write( "So\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "to n factors\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\"M%5En+=+P%2AD%5En%2AP%5E%28-1%29\"\r\n" );
document.write( "\r\n" );
document.write( "Any power of a diagonal matrix is the matrix whose elements are that\r\n" );
document.write( "power of the elements, so we have the final answer as:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );