Chapter 1.2 MatrixAlgebra

1 a b b a addition is commutative 2 a b c

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Unformatted text preview: , µ ∈ R. 1 A + B = B + A (addition is commutative). 2 (A + B ) + C = A + (B + C ) (addition is associative). 3 (λµ)A = λ(µA) (scalar multiplication is associative). 4 (λ + µ)A = λA + µA (matrices distribute through scalar multiplication). 5 λ(A + B ) = λA + λB (scalar multiplication distributes through matrix addition). Outline Matrices Matrix Algebra Matrix Equations Trace and Transpose Properties of Addition and Scalar Multiplication Theorem The following equations hold for any m × n matrices A, B, and C , and scalars λ, µ ∈ R. 1 A + 0 = A = 0 + A where 0 is the n × m zero matrix (0 is the additive identity). 2 A − A = 0 (A and −A are additive inverses). 3 0A = 0 = A0 where 0A represents scalar multiplication and 0 represents the n × m zero matrix. Proof. Use properties of real numbers applied entry by entry. Outline Matrices Matrix Algebra Matrix Equations Trace and Transpose The Product of a Matrix and a Column Vector Let A = (aij ) be an m × n matrix hav...
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This note was uploaded on 02/10/2014 for the course MATH 2270 taught by Professor Kenyonj.platt during the Spring '14 term at Snow College.

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