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Linear Algebra (wi1403lr) Lecture no.6 EWI / DIAM / Numerical Analysis group Matthias Möller 02/05/2014 M. Möller (EWI/NA group) LA (wi1403lr) 02/05/2014 1 / 14 Review of lecture no.5 2.1 Matrix operations • sums and scalar multiples of matrices • matrix-matrix multiplication • powers of a matrix • transpose of a matrix M. Möller (EWI/NA group) LA (wi1403lr) 02/05/2014 2 / 14 Learning objectives of lecture no.6 You will learn • to compute the inverse of a quadratic matrix (if it exists) • to decide whether a matrix is invertible • to compute the determinant of a quadratic matrix • to use the concept of invertible matrices to compute the inverse transformation from the codomain back to the domain M. Möller (EWI/NA group) LA (wi1403lr) 02/05/2014 3 / 14 Multiplicative inverse Let a be an arbitrary scalar. The formal way to consider the division operation 1/a is as follows: 1 Definition: We seek the multiplicative inverse of a such that a−1 · a = 1 and a · a−1 = 1 2 Existence: The notation for it is a−1 and it exists if a 6= 0. 3 Uniqueness: The multiplicative inverse is unique. M. Möller (EWI/NA group) LA (wi1403lr) 02/05/2014 4 / 14 Inverse of a matrix Let A be an ’arbitrary’ n × n matrix. 1 Definition: 2 Existence: 3 Uniqueness: M. Möller (EWI/NA group) LA (wi1403lr) 02/05/2014 5 / 14 Inverse of a matrix Let A be an ’arbitrary’ n × n matrix. 1 Definition: We say that A is invertible if there exists an n × n matrix A−1 such that A−1 A = I and AA−1 = I with I being the n × n identity matrix. If such inverse matrix does not exist then we call A a singular matrix. 2 Existence: 3 Uniqueness: M. Möller (EWI/NA group) LA (wi1403lr) 02/05/2014 5 / 14 Inverse of a matrix Let A be an ’arbitrary’ n × n matrix. 1 Definition: We say that A is invertible if there exists an n × n matrix A−1 such that A−1 A = I and AA−1 = I with I being the n × n identity matrix. If such inverse matrix does not exist then we call A a singular matrix. 2 Existence: 3 Uniqueness: The inverse matrix is unique. If there was another inverse Ã−1 then Ã−1 = Ã−1 I = Ã−1 (AA−1 ) = (Ã−1 A)A−1 = IA−1 = A−1 M. Möller (EWI/NA group) LA (wi1403lr) 02/05/2014 5 / 14 Inverse of a matrix Let A be an ’arbitrary’ n × n matrix. 1 Definition: We say that A is invertible if there exists an n × n matrix A−1 such that A−1 A = I and AA−1 = I with I being the n × n identity matrix. If such inverse matrix does not exist then we call A a singular matrix. 2 Existence: ??? 3 Uniqueness: The inverse matrix is unique. If there was another inverse Ã−1 then Ã−1 = Ã−1 I = Ã−1 (AA−1 ) = (Ã−1 A)A−1 = IA−1 = A−1 M. Möller (EWI/NA group) LA (wi1403lr) 02/05/2014 5 / 14 Computing the inverse of a 2 × 2 matrix a b Let A = . If ad − bc 6= 0, then A is invertible and c d −1 A 1 d −b = ad − bc −c a If ad − bc = 0, then A is not invertible. This quantity is termed the determinant of matrix A det A = ad − bc Decide if the inverse of matrix A = 1 2 exists and compute it if 5 12 this is possible. M. Möller (EWI/NA group) LA (wi1403lr) 02/05/2014 6 / 14 Take home lesson a b Let A = be an ’arbitrary’ 2 × 2 matrix. c d 1 Definition: The inverse matrix is defined as 1 d −b −1 A = ad − bc −c a provided that ad − bc 6= 0. 2 Existence: The inverse matrix exists if and only if det A = ad − bc 6= 0 3 Uniqueness: The inverse matrix is unique. M. Möller (EWI/NA group) LA (wi1403lr) 02/05/2014 7 / 14 Solution of linear equations If the n × n matrix A is invertible, then for each vector b in Rn , the equation Ax = b has the unique solution x = A−1 b. 1 2 6 −1 −1 Let A = with inverse matrix A = . 5 12 −2.5 0.5 Solve the linear equations −1 Ax = 3 M. Möller (EWI/NA group) 1 Ax = −5 LA (wi1403lr) 2 Ax = 6 02/05/2014 8 / 14 Solution of linear equations If the n × n matrix A is invertible, then for each vector b in Rn , the equation Ax = b has the unique solution x = A−1 b. 1 2 6 −1 −1 Let A = with inverse matrix A = . 5 12 −2.5 0.5 Solve the linear equations −1 Ax = 3 1 Ax = −5 2 Ax = 6 There must be a less tedious way to do this! M. Möller (EWI/NA group) LA (wi1403lr) 02/05/2014 8 / 14 Draft of a solution idea Write your problem(s) into augmented form A b1 b2 b3 and transform your matrix A into the 2 × 2 identity matrix by applying only elementary row operations: 1 Add/subtract the multiple of one row to another row. 2 Scale one row by a scalar. 3 Swap the position of two rows. M. Möller (EWI/NA group) LA (wi1403lr) 02/05/2014 9 / 14 Draft of a solution idea Write your problem(s) into augmented form A b1 b2 b3 and transform your matrix A into the 2 × 2 identity matrix by applying only elementary row operations: 1 Add/subtract the multiple of one row to another row. 2 Scale one row by a scalar. 3 Swap the position of two rows. We found a simple algorithm to solve a sequence of linear equations with the same matrix A and arbitrarily many right-hand sides bi . M. Möller (EWI/NA group) LA (wi1403lr) 02/05/2014 9 / 14 Computing the inverse of a matrix 1 2 1 0 Let A = . Solve the linear equations Ax = and Ax = 5 12 0 1 by writing it in augmented form and applying elementary operations. M. Möller (EWI/NA group) LA (wi1403lr) 02/05/2014 10 / 14 Computing the inverse of a matrix 1 2 1 0 Let A = . Solve the linear equations Ax = and Ax = 5 12 0 1 by writing it in augmented form and applying elementary operations. 1 2 | 1 0 1 0 | 6 −1 ∼ ··· ∼ 5 12 | 0 1 0 12 | −2.5 0.5 M. Möller (EWI/NA group) LA (wi1403lr) 02/05/2014 10 / 14 Computing the inverse of a matrix 1 2 1 0 Let A = . Solve the linear equations Ax = and Ax = 5 12 0 1 by writing it in augmented form and applying elementary operations. 1 2 | 1 0 1 0 | 6 −1 ∼ ··· ∼ 5 12 | 0 1 0 12 | −2.5 0.5 An n × n matrix A is invertible if and only if A is row equivalent to I , and in this case, any sequence of elementary row operations that reduces A to I also transforms I into A−1 . M. Möller (EWI/NA group) LA (wi1403lr) 02/05/2014 10 / 14 Algorithm for finding A−1 1 Write down the augmented matrix A I 2 Row reduce the augmented matrix to I A−1 by applying only elementary row operations. If the algorithm breaks down before A has been transformed (i.e. row reduced) into I , then A does not have an inverse. M. Möller (EWI/NA group) LA (wi1403lr) 02/05/2014 11 / 14 Properties of the inverse of a matrix • If A is an invertible matrix, then A−1 is also invertible and (A−1 )−1 = A • If A and B are n × n invertible matrices, then so is AB, and (AB)−1 = B −1 A−1 • If A is an invertible matrix, then so is AT , and (AT )−1 = (A−1 )T Proofs of each statement can be found in the book. M. Möller (EWI/NA group) LA (wi1403lr) 02/05/2014 12 / 14 The invertible matrix theorem Let A be a square n × n matrix. Then the following statements are are either all true all false, that is, they are equivalent. a. A is an invertible matrix. ... l. AT is an invertible matrix. Proof (by showing equivalence of the two statements) M. Möller (EWI/NA group) LA (wi1403lr) 02/05/2014 13 / 14 The invertible matrix theorem Let A be a square n × n matrix. Then the following statements are are either all true all false, that is, they are equivalent. a. A is an invertible matrix. ... l. AT is an invertible matrix. Proof (by showing equivalence of the two statements) • (a) ⇒ (l): By the third property on the previous slide. M. Möller (EWI/NA group) LA (wi1403lr) 02/05/2014 13 / 14 The invertible matrix theorem Let A be a square n × n matrix. Then the following statements are are either all true all false, that is, they are equivalent. a. A is an invertible matrix. ... l. AT is an invertible matrix. Proof (by showing equivalence of the two statements) • (a) ⇒ (l): By the third property on the previous slide. • (l) ⇒ (a): By the third property on the previous slide with A and AT interchanged. M. Möller (EWI/NA group) LA (wi1403lr) 02/05/2014 13 / 14 The invertible matrix theorem Let A be a square n × n matrix. Then the following statements are are either all true all false, that is, they are equivalent. a. A is an invertible matrix. ... l. AT is an invertible matrix. Proof (by showing equivalence of the two statements) • (a) ⇒ (l): By the third property on the previous slide. • (l) ⇒ (a): By the third property on the previous slide with A and AT interchanged. • This implication in both directions proves that both statements are equivalen, i.e. (a) ⇔ (l). M. Möller (EWI/NA group) LA (wi1403lr) 02/05/2014 13 / 14 The invertible matrix theorem Let A be a square n × n matrix. Then the following statements are are either all true all false, that is, they are equivalent. a. A is an invertible matrix. j. There is an n × n matrix C such that CA = I . d. The equation Ax = 0 has only the trivial solution. c. A has n pivot positions. b. A is row equivalent to the n × n identity matrix. Proof (by establishing a circle of implications) (a) ⇒ (j) ⇒ (d) ⇒ (c) ⇒ (b) ⇒ (a) M. Möller (EWI/NA group) LA (wi1403lr) 02/05/2014 14 / 14