203 - Properties: For n = 1, the definition reduces to the...

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1 Properties: For n = 1, the definition reduces to the multiplicative inverse ( ab = ba = 1). If B is an inverse of A , then A is an inverse of B , i.e., A and B are inverses to each other. Example: Example: Some matrices have no inverse, like O R n × n and since Proof Suppose B and C are both inverses of A . Then Notation: The inverse of A , if exists, is denoted by A 1 . Symbolically, the inverse may be used to solve matrix equations: A x = b However, this method is computationally inefficient.
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2 Example: Proof (a) A 1 and A are the inverse to each other. (b) ( AB )( B 1 A 1 ) = A ( BB 1 ) A 1 = AI n A 1 = AA 1 = I n ; similarly, ( B 1 A 1 ) ( AB ) = I n . (c) A T ( A 1 ) T = ( A 1 A ) T = ( I n ) T = I n ; similarly, ( A 1 ) T A T = I n . then A T is also invertible, and ( A T ) 1 = ( A 1 ) T . 2 Definition. An n × n matrix E is called an elementary matrix if E can be obtained from I n by a single elementary row operation. . 1 0 2 0 1 0 0 0 1 , 1 0 0 0 4 0 0 0 1 , 0 1 0 1 0 0 0 0 1 : Example 3 2 1 = = = E E E . 10 7 4 3 2 1 6 5 4 3 2 1 1 0 2 0 1 0 0 0 1 6 5 4 3 2 1 : Example 3 =
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203 - Properties: For n = 1, the definition reduces to the...

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