document.write( "Question 1182268: Suppose A,B and AB _1 are non-degerate and have same order prove that
\n" ); document.write( "(A-B¹) is non-degerate;
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Algebra.Com's Answer #812517 by ikleyn(52847)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "Your writing in your post is INCORRECT,\r
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\n" ); document.write( "\n" ); document.write( "THEREFORE,  I edited it in couple of lines below.\r
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document.write( "          Suppose A,B and AB-I are non-degenerate and have same order. Prove that\r\n" );
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document.write( "             \"%28A-B%5E%28-1%29%29\"  is non-degenerate;\r\n" );
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document.write( "             \"%28A-B%5E%28-1%29%29%5E%28-1%29\" is non-degenerate. Find the inverse of this matrix \r\n" );
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\n" ); document.write( "\n" ); document.write( "This problem was posted to the forum several years ago,  and I solved it under this link\r
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\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/Matrices-and-determiminant/Matrices-and-determiminant.faq.question.1099395.html\r
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\n" ); document.write( "\n" ); document.write( "Below I copy-pasted that solution here again for your convenience.\r
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document.write( "    Since the matrix B is non-degenerated, it has the inverse matrix  \"B%5E%28-1%29\".\r\n" );
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document.write( "    Multiply the matrix \"A%2AB-I\" by the matrix \"B%5E%28-1%29\"  on the right. You will get\r\n" );
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document.write( "    \"%28A%2AB-I%29%2AB%5E%28-1%29\" = \"A+-+B%5E%28-1%29\".\r\n" );
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document.write( "    Since both matrices on the left side of the equation are non-degenerated, their product on the right side is non-degenerated, too.\r\n" );
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document.write( "    QED\r\n" );
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