u2 - Unit II Matrix Algebra 1 Algebraic Operations with...

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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 = 0 , 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 . This property of having unique solutions characterizes invertible matrices. 1
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Theorem 1. The following are equivalent, for an n × n matrix A : (1) A is invertible, (2) A is row-equivalent to the identity matrix, (3) the only solution of A x = 0 is x = 0 , (4) for any b R n , the system A x = b has a solution.
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