document.write( "Question 1164621: Suppose that A is a square matrix and A4 = 0 (the zero matrix). Show that (10 points)
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Algebra.Com's Answer #852053 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Suppose that A is a square matrix and $\"A%5E4\"$ = 0 (the zero matrix). Show that
\n" ); document.write( " $\"%28I+-+A%29%5E%28-1%29\"$ = $\"I\"$ + $\"A\"$ + $\"A%5E2\"$ + $\"A%5E3\"$ .
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document.write( "Let  B  be the matrix   +  +  + .\r\n" );
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document.write( "We need to show that  (I-A)*B = I  and  B*((I-A) = I.\r\n" );
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document.write( "For the first equality, we have\r\n" );
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document.write( "    (I-A)*B =  =  = \r\n" );
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document.write( "                                        =  -  =  = I.\r\n" );
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document.write( "    Here we used the fact that   = 0,  which is given.\r\n" );
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document.write( "    Thus, the property  (I-A)*B = I  is proven.\r\n" );
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document.write( "For the second equality, we have\r\n" );
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document.write( "    B*(I-A) =  =  -  = \r\n" );
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document.write( "                                        =  -  =  = I.\r\n" );
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document.write( "    Again, here we used the fact that   = 0,  which is given.\r\n" );
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document.write( "    Thus, the property  B*(I-A) = I  is proven.\r\n" );
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