# Appendix_SC - Appendix SC SC.1 Definitions Matrices A...

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1 Appendix SC Matrices SC.1 Definitions A matrix is defined as a rectangular array of elements, usually numbers, which are arranged in m rows and n columns. Thus: A [] = = mn m m n n mn ij a a a a a a a a a a ... ... ... 2 1 2 22 21 1 12 11 M M M M (SC.1.1) The transpose of an m × n matrix A is an n × m matrix A T in which the rows and columns of A are interchanged. The element ij a of A becomes ji a of A T . A row matrix has 1 = m , in which case, A [ ] n a a a 1 12 11 ... = . In a column matrix , 1 = n , so that A = 1 21 11 m a a a M . It follows that the transpose of a row matrix is a column matrix, and conversely. If n m = , a square matrix results. An identity matrix is a square matrix in which the elements along the main diagonal are all equal to 1 ) for 1 ( j i a ij = = , with all the other elements zero ) for 0 ( j i a ij = . The determinant of the elements of a square matrix is denoted by det A . The adjoint matrix of a square matrix A of order n × n is the transpose of the matrix formed by the cofactors of each element in the determinant of A . Formally: adj A [ ] T ij Δ = (SC.1.2) where ij Δ is the cofactor of ij a (Section SA.2). These definitions are considered in the following example. ___________________________________________________________ Example SC.1.1 Given A = 1 2 1 2 1 3 1 3 1 . Find adj A and det A . Solution : the cofactors of the elements of A are: 5 ) 4 1 ( 11 = = Δ , 5 ) 2 3 ( 12 = = Δ ,

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2 5 ) 1 6 ( 13 = = Δ , 1 ) 2 3 ( 21 = + = Δ , 0 ) 1 1 ( 22 = + = Δ , 1 ) 3 2 ( 23 = = Δ , 7 ) 1 6 ( 31 = + = Δ , 5 ) 3 2 ( 32 = + = Δ , 8 ) 9 1 ( 33 = = Δ . The matrix of the cofactors is: B = 8 5 7 1 0 1 5 5 5 . The transpose of this matrix is: adj A = 8 1 5 5 0 5 7 1 5 . The determinant of A is: 5 ) 5 ( 1 ) 5 ( 3 ) 5 ( 1 = + . ___________________________________________________________ SC.2 Matrix Operations When two m × n matrices A and B are added, or subtracted, the result is a matrix C whose elements are given by: ij ij ij b a c ± = for all i and j (SC.2.1) In other words the elements of the two matrices are simply added, or subtracted, element by element. When a matrix is multiplied, or divided, by a scalar k , each element is multiplied, or divided, by k . Two matrices A and B can be multiplied together to form a product matrix C if and only if the number of columns of A is equal to the number of rows of B . Thus, if A is an m × s matrix and A is an s × n matrix, then C is an m × n matrix whose elements are given by: kj ik s k k ij b a c = = = 1 for m i ..., , 1 = and n j ..., , 1 = (SC.2.2) In words, the element in the i th row and j th column of C is obtained by taking the i th row of A and the j th column of B , each of which contains s elements, multiplying them element by element and adding the products. These operations are illustrated by the following example.
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## This note was uploaded on 02/17/2011 for the course EECE 210 taught by Professor Riadchedid during the Fall '07 term at American University of Beirut.

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Appendix_SC - Appendix SC SC.1 Definitions Matrices A...

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