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Unformatted text preview: Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Algebraic Rules (Cont’d) Proof of (3): Goal Prove A(B + C ) = AB + AC Let A = (aij )m×n , B = (bij )n×r and C = (cij )n×r . Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Algebraic Rules (Cont’d) Proof of (3): Goal Prove A(B + C ) = AB + AC Let A = (aij )m×n , B = (bij )n×r and C = (cij )n×r . Since B + C = (bij + cij )n×r , A(B + C ) = (dij )m×r where
n n n dij =
k =1 aik (bkj + ckj ) =
k =1 aik bkj +
k =1 aik ckj . Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Algebraic Rules (Cont’d) Proof of (3): Goal Prove A(B + C ) = AB + AC Let A = (aij )m×n , B = (bij )n×r and C = (cij )n×r . Since B + C = (bij + cij )n×r , A(B + C ) = (dij )m×r where
n n n dij =
k =1 aik (bkj + ckj ) =
k =1 aik bkj +
k =1 aik ckj . On the other hand,
AB = (
n ab) k=1 ik kj m×r n ab k=1 ik kj and AC = (
+ n ac) k=1 ik kj m×r . Thus, AB + AC = n ac k=1 ik kj m×r = A(B + C ). Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Algebraic Rules (Cont’d) Proof of (5): Goal Prove A(BC ) = (AB)C Let A = (aij )m×n , B = (bij )n×r , C = (cij )r×s . Check dimension of A(BC ) = dimension of (AB)C . Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Algebraic Rules (Cont’d) Proof of (5): Goal Prove A(BC ) = (AB)C Let A = (aij )m×n , B = (bij )n×r , C = (cij )r×s . Check dimension of A(BC ) = dimension of (AB)C .
• The dimension of BC is n × s, so A(BC ) is of dim m × s. • The dimension of AB is m × r , so (AB)C is of dim m × s. Remains to check the corresponding entries are equal. Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Algebraic Rules (Cont’d) Reminder Checked: Both A(BC ) and (AB)C are of dimension m × s where A = (aij )m×n , B = (bij )n×r , C = (cij )r ×s .
r Let BC = (ekj )n×s . Then ekj =
l =1 n bkl clj
n r The (i, j)th entry of A(BC ) is
k =1 aik ekj =
k =1 aik
l =1 bkl clj Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Algebraic Rules (Cont’d) Reminder Checked: Both A(BC ) and (AB)C are of dimension m × s where A = (aij )m×n , B = (bij )n×r , C = (cij )r ×s .
r Let BC = (ekj )n×s . Then ekj =
l =1 n bkl clj
n r The (i, j)th entry of A(BC ) is
k =1 n aik ekj =
k =1 aik
l =1 bkl clj Let AB = (dil )m×r . Then dil =
k =1 r aik bkl .
r n The (i, j)th entry of (AB)C is
l =1 dil clj =
l =1 k =1 aik bkl clj Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Algebraic Rules (Cont’d) Reminder Checked: Both A(BC ) and (AB)C are of dimension m × s where A = (aij )m×n , B = (bij )n×r , C = (cij )r ×s .
r Let BC = (ekj )n×s . Then ekj =
l =1 n bkl clj
n r The (i, j)th entry of A(BC ) is
k =1 n aik ekj =
k =1 aik
l =1 bkl clj Let AB = (dil )m×r . Then dil =
k =1 r aik bkl .
r n The (i, j)th entry of (AB)C is
l =1 dil clj =
l =1 k =1 aik bkl clj Observe that the two sums equal each other. Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Power of a Matrix & Identity Matrices Deﬁnition Let k be a natural number, and A be a square matrix. Deﬁne
Ak := AA · · · A
k times Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Power of a Matrix & Identity Matrices Deﬁnition Let k be a natural number, and A be a square matrix. Deﬁne
Ak := AA · · · A
k times Deﬁnition The n × n identity matrix is the square matrix
I = (δij ) where δij = 1 0 if i = j, if i = j. Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Power of a Matrix & Identity Matrices Deﬁnition Let k be a natural number, and A be a square matrix. Deﬁne
Ak := AA · · · A
k times Deﬁnition The n × n identity matrix is the square matrix
I = (δij ) where δij = 1 0 if i = j, if i = j. Property An identity matrix behaves like the number 1. Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Homework 8 Reading Leon (7th edition): p.41  p.44, p.47  49 Leon (8th edition): p.44  p.47, p.50  52 Homework 8 Leon (7th edition): Chapter 1 Section 3 Qn. 1012, 2023. Leon (8th edition): Section 1.4 Qn. 14, 79. Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Identity Matrices Example. Let I be the 3 × 3 identity matrix, i.e. 1 0 0 I = 0 1 0. 001 3 4 1 Evaluate IA and AI where A = 2 6 3. 018 Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Identity Matrices Example. Let I be the 3 × 3 identity matrix, i.e. 1 0 0 I = 0 1 0. 001 3 4 1 Evaluate IA and AI where A = 2 6 3. 018 Example. Let ei be the ith column of the n × n identity matrix. Suppose A is an m × n matrix. What is Aei ? Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Matrix Inversion  Motivation Warning
Matrix multiplication is not commutative. i.e. AB = BA in general. The concept of division is not introduced. Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Matrix Inversion  Motivation Warning
Matrix multiplication is not commutative. i.e. AB = BA in general. The concept of division is not introduced. Question: What is division? Ans. 6 ÷ 2 = 3 or
6 = 3 means 6 = 2 × 3 = 3 × 2. 2 Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Matrix Inversion  Motivation Warning
Matrix multiplication is not commutative. i.e. AB = BA in general. The concept of division is not introduced. Question: What is division?
6 = 3 means 6 = 2 × 3 = 3 × 2. 2 However, AB may not equal BA in matrices! Ans. 6 ÷ 2 = 3 or Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Matrix Inversion  Motivation Warning
Matrix multiplication is not commutative. i.e. AB = BA in general. The concept of division is not introduced. Question: What is division?
6 = 3 means 6 = 2 × 3 = 3 × 2. 2 However, AB may not equal BA in matrices! View 6 as 6 × 2−1 . 2 Ans. 6 ÷ 2 = 3 or Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Matrix Inversion  Motivation Warning
Matrix multiplication is not commutative. i.e. AB = BA in general. The concept of division is not introduced. Question: What is division?
6 = 3 means 6 = 2 × 3 = 3 × 2. 2 However, AB may not equal BA in matrices! View 6 as 6 × 2−1 . 2 Ans. 6 ÷ 2 = 3 or Question: Can we deﬁne a concept like reciprocal for matrices? Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Matrix Inversion  Motivation Warning
Matrix multiplication is not commutative. i.e. AB = BA in general. The concept of division is not introduced. Question: What is division?
6 = 3 means 6 = 2 × 3 = 3 × 2. 2 However, AB may not equal BA in matrices! View 6 as 6 × 2−1 . 2 Ans. 6 ÷ 2 = 3 or Question: Can we deﬁne a concept like reciprocal for matrices? What is reciprocal of a number? Ans. 2−1 is a number for which 2 × 2−1 = 1 = 2−1 × 2. Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Matrix Inversion Deﬁnition An n × n matrix A is nonsingular or invertible if there exists a matrix B such that
AB = I = BA where I is the identity matrix of order n. Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Matrix Inversion Deﬁnition An n × n matrix A is nonsingular or invertible if there exists a matrix B such that
AB = I = BA where I is the identity matrix of order n. We call B a multiplicative inverse of A. Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Matrix Inversion Deﬁnition An n × n matrix A is nonsingular or invertible if there exists a matrix B such that
AB = I = BA where I is the identity matrix of order n. We call B a multiplicative inverse of A. Remark Inverse of a matrix is unique. Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Matrix Inversion Deﬁnition An n × n matrix A is nonsingular or invertible if there exists a matrix B such that
AB = I = BA where I is the identity matrix of order n. We call B a multiplicative inverse of A. Remark Inverse of a matrix is unique.
Question How to prove? Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra Matrix Inversion Deﬁnition An n × n matrix A is nonsingular or invertible if there exists a matrix B such that
AB = I = BA where I is the identity matrix of order n. We call B a multiplicative inverse of A. Remark Inverse of a matrix is unique.
Question How to prove? Notation Denote the inverse of A by A−1 , if A is nonsingular Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra −1 1 2 3 0 1 4 001 Matrix Inversion  Example 1 −2 5 0 1 −4. 00 1 Example. Check that is Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra −1 1 2 3 0 1 4 001 Matrix Inversion  Example 1 −2 5 0 1 −4. 00 1 Example. Check that is 1 0 Example. Is A = invertible? 00 Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra −1 1 2 3 0 1 4 001 Matrix Inversion  Example 1 −2 5 0 1 −4. 00 1 Example. Check that is 1 0 Example. Is A = invertible? 00 Remark In numbers only 0 has no inverse, but in matrices there are nonzero noninvertible matrices. Chapter 1. Matrices and Systems of Equations Math1111 Matrix Algebra −1 1 2 3 0 1 4 001 Matrix Inversion  Example 1 −2 5 0 1 −4. 00 1 Example. Check that is 1 0 Example. Is A = invertible? 00 Remark In numbers only 0 has no inverse, but in matrices there are nonzero noninvertible matrices. Remark An n × n matrix is said to be singular if it has no inverse. ...
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This document was uploaded on 05/04/2011.
 Spring '11

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