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Unformatted text preview: Section 2.9 Transposes
Definition: The transpose of A, T denoted A , is found by interchanging the rows and columns of A. 1 2 3 A= 4 5 6 Example: Let T find A .
1 4 AT = 2 5 3 6 Note that if A is mXn then AT is nXm. Properties (Theorem): If A & B are mXn matrices and C is nXm, then (a) (A + B)T = AT + BT (b) (AC)T = CTAT Note that here AC is mXm. Since CT is mXn and AT is nXm, then CTAT is mXm. ATCT would be nXn, the "wrong" size. (c) (AT)T = A (d) If A is square, then T det(A) = det (A ) (e) If A is square and invertible, 1 T T 1 then (A ) = (A ) Example: Find (A ) if 1 0 2 A 1 = 4  1 1 3 2 2 T 1 Since (AT)1 = (A1)T , the easiest way to do this is transpose A1. (A ) 1 T 1 4 3 = 0  1 2 2 1 2 Definition: A symmetric matrix is one where A = AT. Example: The following matrices are symmetric. 1 2 3 A = 2 4 5 3 5 6 1 2 3 4 2 5 6 8 B= 3 6 7 9 4 8 9 0 ...
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 Fall '08
 JHENGLAND
 Linear Algebra, Algebra

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