document.write( "Question 1161345: Let A= [−47 −90 ; 27 52].
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Algebra.Com's Answer #784993 by Edwin McCravy(20065)\"\" \"About 
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Let A= [−47 −90 ; 27 52].
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document.write( "We diagnalize the matrix:\r\n" );
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document.write( "\"matrix%281%2C3%2CA%2C%22%22=%22%22%2C%0D%0A%28matrix%282%2C2%2C-47%2C-90%2C27%2C52%29%29%29\"\r\n" );
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document.write( "We find the eigenvalues\r\n" );
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document.write( "\"abs%28matrix%282%2C2%2C-47-lambda%2C-90%2C27%2C52-lambda%29%29%29\"\"%22%22=%22%22\"\"0\"\r\n" );
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document.write( "\"%28-47-lambda%29%2852-lambda%29-%28-90%29%2827%29=0\"\r\n" );
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document.write( "\"-2444-5lambda%2Blambda%5E2%2B2430=0\"\r\n" );
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document.write( "\"lambda%5E2-5lambda-14=0\"\r\n" );
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document.write( "\"%28lambda-7%29%28lambda%2B2%29=0\"\r\n" );
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document.write( " λ-7=0;  λ+2=0   \r\n" );
document.write( "   λ=7;   λ=-2\r\n" );
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document.write( "by writing A as \r\n" );
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document.write( "\"matrix%281%2C3%2CA%2C%22%22=%22%22%2CSDS%5E%28-1%29%29\"\r\n" );
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document.write( "where D is the diagonal matrix with the two eigenvalues on the \r\n" );
document.write( "main diagonal:\r\n" );
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document.write( "\"D+=+%28matrix%282%2C2%2C7%2C0%2C0%2C-2%29%29\"\r\n" );
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document.write( "and the matrix S is \r\n" );
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document.write( "\"S=%28matrix%281%2C2%2CV%5B1%5D%2CV%5B2%5D%29%29\"\r\n" );
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document.write( "where the V's are the two column eigenvectors for the two eigenvalues\r\n" );
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document.write( "We find V1 which is the eigengvector for the eigenvalue λ=7.\r\n" );
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document.write( "We find solutions for\r\n" );
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document.write( "\"%28A-7I%29X=0\"\r\n" );
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document.write( "\"-54x%5B1%5D-90x%5B2%5D=0\"\r\n" );
document.write( "Divide thru by -18\r\n" );
document.write( "\"3x%5B1%5D%2B5x%5B2%5D=0\"\r\n" );
document.write( "\"3x%5B1%5D=-5x%5B2%5D\"\r\n" );
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document.write( "We can take x1=1 and x1=1\r\n" );
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document.write( "So \r\n" );
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document.write( "\"v%5B1%5D=%28matrix%282%2C1%2C-5%2C3%29%29\"\r\n" );
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document.write( "Now we do the same for the other eigenvalue\r\n" );
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document.write( "We find solutions for\r\n" );
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document.write( "\"%28A%2B2I%29X=0\"\r\n" );
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document.write( "\"-45x%5B1%5D-90x%5B2%5D=0\"\r\n" );
document.write( "Divide thru by -45\r\n" );
document.write( "\"x%5B1%5D%2B2x%5B2%5D=0\"\r\n" );
document.write( "\"x%5B1%5D=-2x%5B2%5D\"\r\n" );
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document.write( "We can take x1=1 and x2=1\r\n" );
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document.write( "So \r\n" );
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document.write( "\"v%5B2%5D=%28matrix%282%2C1%2C-2%2C1%29%29\"\r\n" );
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document.write( "So\r\n" );
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document.write( "\"S=%28matrix%282%2C2%2C-5%2C3%2C-2%2C1%29%29\"\r\n" );
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document.write( "And since the determinant of S is 1, to find S-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" );
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document.write( "\"S%5E%28-1%29=%28matrix%282%2C2%2C1%2C2%2C-3%2C-5%29%29\"\r\n" );
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document.write( "Then \r\n" );
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document.write( "\"A=SDS%5E%28-1%29\"\r\n" );
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document.write( "Edwin
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