jerri-2-9 - Section 2.9 Transposes Definition: The...

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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 = (A-1)T , the easiest way to do this is transpose A-1. (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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