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Lect5 - Chapter 1 Matrices and Systems of Equations...

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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 = ( a ij ) m × n , B = ( b ij ) n × r and C = ( c ij ) n × r .
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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 = ( a ij ) m × n , B = ( b ij ) n × r and C = ( c ij ) n × r . Since B + C = ( b ij + c ij ) n × r , A ( B + C ) = ( d ij ) m × r where d ij = n X k = 1 a ik ( b kj + c kj ) = n X k = 1 a ik b kj + n X k = 1 a ik c kj .
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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 = ( a ij ) m × n , B = ( b ij ) n × r and C = ( c ij ) n × r . Since B + C = ( b ij + c ij ) n × r , A ( B + C ) = ( d ij ) m × r where d ij = n X k = 1 a ik ( b kj + c kj ) = n X k = 1 a ik b kj + n X k = 1 a ik c kj . On the other hand, AB = ( n k = 1 a ik b kj ) m × r and AC = ( n k = 1 a ik c kj ) m × r . Thus, AB + AC = n k = 1 a ik b kj + n k = 1 a ik c kj m × r = A ( B + C ) .
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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 = ( a ij ) m × n , B = ( b ij ) n × r , C = ( c ij ) r × s . Check dimension of A ( BC ) = dimension of ( AB ) C .
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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 = ( a ij ) m × n , B = ( b ij ) n × r , C = ( c ij ) 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.
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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 = ( a ij ) m × n , B = ( b ij ) n × r , C = ( c ij ) r × s . Let BC = ( e kj ) n × s . Then e kj = r X l = 1 b kl c lj The ( i , j ) th entry of A ( BC ) is n X k = 1 a ik e kj = n X k = 1 a ik r X l = 1 b kl c lj
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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 = ( a ij ) m × n , B = ( b ij ) n × r , C = ( c ij ) r × s . Let BC = ( e kj ) n × s . Then e kj = r X l = 1 b kl c lj The ( i , j ) th entry of A ( BC ) is n X k = 1 a ik e kj = n X k = 1 a ik r X l = 1 b kl c lj Let AB = ( d il ) m × r . Then d il = n X k = 1 a ik b kl .
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