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Unformatted text preview: Unit II: Matrix Algebra 1. Algebraic Operations with Matrices addition of matrices, multiplication of a matrix by a scalar Definition of product of matrices: If A is an m n matrix, and B is an n p matrix with columns b 1 ,..., b p R n , then AB is the m p matrix with columns A b 1 ,...,A b p . In other words, AB = A [ b 1 ,..., b p ] = [ A b 1 ,...,A b p ] . Lemma. If A is an m n matrix, and B is an n p matrix, then ( AB ) ik = n X j =1 A ij B jk , 1 i m, 1 k p. Proofsketch. Use the definition of A x with x replaced by the k th column of B . Rules for operations on matrices: matrix addition is associative and commutative, ma- trix multiplication is associative, and the usual distributive rules hold. However, matrix multiplication is NOT commutative. vectors as matrices, row vectors, column vectors, vector form of a system of linear equa- tions, A x = , A x = b identity matrix, diagonal matrix, upper triangular matrix, lower triangular matrix transpose A t of a matrix A , transpose of a product of matrices, ( AB ) t = B t A t , symmetric matrix ( A t = A ) 2. Invertible Matrices An n n matrix A is invertible if there is a matrix B such that AB = BA = I . The matrix B is called the inverse of A and denoted by A- 1 . The next lemma shows that the inverse B of A is unique, so that A- 1 is well defined. Lemma. If BA = I , and AC = I , then B = C . Proof. B = BI = B ( AC ) = ( BA ) C = IC = C . Lemma. If the square matrix A is invertible, then A- 1 is invertible, and ( A- 1 )- 1 = A . Proof. This follows from AA- 1 = I = A- 1 A . Lemma. If the n n matrices A and B are invertible, then AB is invertible, and ( AB )- 1 = B- 1 A- 1 . Lemma. An n n matrix A is invertible if and only if its transpose A t is invertible, and ( A t )- 1 = ( A- 1 ) t . Proofsketch . Use ( AB ) t = B t A t and ( AB )- 1 = B- 1 A- 1 . If A is invertible, then the system A x = b has a unique solution, namely, x = A- 1 b ....
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