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Section 5.3 MATH 166:503 April 14, 2015 Topics from last notes: matrix, order, row and column matrices, square matrix, multiplication by a scalar, addition of matrices, transpose of a matrix, zero matrix, properties of matrix addition, using matrix addition to solve word problems, matrix multiplication, properties of matrix multiplication, identity matrix, using matrix multiplication to solve word problems 4 SYSTEMS OF LINEAR EQUATIONS AND MODELS ex. (from WebAssign) The matrix below is in row-reduced form x M= 1 0 y 0 1 Solve the system. (x, y, z, u) = 1 z 0 6 u 0 4 7 7 5 MATRICES 5.3 Inverse of a Square Matrix A n x n matrix B is the inverse of an n x n matrix A if AB = BA = In . If the inverse exists, we denote B = A−1 . To find A−1 : 1. Form the augmented matrix [A|I] 2. Row reduce to get [I|B] if possible 3. Then B = A−1 ex. Find the inverse of A = 1 2 3 4 . Check AA−1 = I. 2 −1 0 8 ex. A = 0 1 1 . Compute A−1 . 3 9 −6 −1 0 8 ex. A = 2 10 1 . Compute A−1 . 3 10 −7 If we write a system of equations as AX = B and A−1 exists, If A−1 does not exist, the system has either no solutions or infinitely many solutions. 3 ex. Solve the following system using matrices 9a + 3b − c =10 1 −a + c =6 2 4a + b =5 ex. Solve the following system 3 2p + q =12 2 4 p + q =14 3 4 ex. Solve the following system 6x1 + 3x2 − x4 =18 x2 + x4 =0 3 x1 − x2 + 4x3 = − 4 2 5