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8. 7. 9. For evaluate m(S). 10. For evaluate p(T). Verify that the diagonal entries of p(T) are p(1), p(– 1), and p(3). 5 TRANSPOSITION If A is an n × m matrix, then the m × n matrix obtained from A by interchanging its rows and columns is called the transpose of A and is denoted by AT. 10 If A is n × m over , then AT is the m × n matrix defined by entij(AT) = entji(A). For example, if then We will find it convenient and space-saving to write column matrices as transposed row matrices. You will see later that transposition is a useful and important operation on matrices which will allow us to deduce column analogs of results about the rows of a matrix and vice versa.

The following theorem lists these properties formally. It is assumed that all matrices written are conformable for addition. 5 Matrix addition has the following properties: I. Commutative: that is, A + B = B + A. II. Associative: that is, A + (B + C) = (A + B) + C. III. There is an identity element for matrix addition. ) IV. There is an additive inverse (negative): that is, for each matrix A, there is a unique matrix – A satisfying A + (– A) = Z. V. Cancellation Law: that is, if A + B = A + C, then B = C.

Now, by the definition of the transpose and the definition of matrix multiplication Thus (AB)T = BTAT as claimed. IV. See Exercise 3. V. AT(A– 1A)T = (A– 1 A)T = IT = I by III and, similarly, (A– 1)TAT = (AA– 1)T = I. Thus (AT)– 1 = (A– 1)T as claimed. The definition of the transpose of a matrix makes perfectly good sense independent of the nature of the elements of the matrix. For matrices whose elements are complex numbers, there is a related idea which turns out to be more useful in practice.

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