Algebraic Properties of Matrix Operations

Algebraic Properties of Matrix Operations
The m x n matrix with all entries of zero is denoted by 푶_풎풏 , for a matrix A of size m x n, we have:

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2. LINEAR ALGEBRA Solving a Matrix Equation a ) 3X + A = B A = Using your skills in matrix addition and scalar multiplication, solve the following matrix equation:   B =   b ) 2 X + 2A = 4 B

3. M117 Linear Algebra Lesson 4 Algebraic Properties of Matrix Operations REPORTERS: Joycee Anne Pura Kristopher Hiloma BSED ─ MATHEMATICS

4. LINEAR ALGEBRA Let A , B and C be m x n matrices. Properties of Matrix Addition A + B = B + A A + ( B + C ) = ( A + B ) + C A =   B =   C =   Commutativity of Addition Associativity of Addition A + B or B + A =   B + C =     Zero Matrix

5. LINEAR ALGEBRA Let A and B be m x n matrices and c and d are scalars. Properties of Scalar Multiplication ( cd ) A = c ( d A ) 1 A = A c ( A + B ) = c A + c B ( c + d ) A = c A + d A Associative Property of Multiplication Multiplicative Identity Distributive Property Distributive Property A =   B =   c = 2 d =  

6. LINEAR ALGEBRA Properties of Zero Matrices A + = A   A =   =     The m x n matrix with all entries of zero is denoted by , for a matrix A of size m x n , we have:   This property states that or the zero matrix is the additive identity for the set of all m x n matrices.  

7. LINEAR ALGEBRA Properties of Zero Matrices A + D =   A = This property states that ￚ A or the negative of A is the additive inverse of A . A + ( ￚ A ) =     ￚ A =   ￚ A = ￚ1(A) The m x n matrix with all entries of zero is denoted by , for a matrix A of size m x n , we have:  

8. LINEAR ALGEBRA Properties of Zero Matrices If c A = , then c = 0 or A = .   A =   c A = A =     c A =   c =   c A = 10   c A =   c = The m x n matrix with all entries of zero is denoted by , for a matrix A of size m x n and a scalar c , we have:  

9. LINEAR ALGEBRA What did you notice? For these first three properties of matrix operations, “ matrices behave like real numbers .” Properties of Matrix Addition Properties of Zero Matrices Commutativity of Addition Associativity of Addition Additive Identity Additive Inverse Zero (0) as a Multiplier Properties of Scalar Multiplication

10. LINEAR ALGEBRA Let A , B and C be matrices with sizes such that the given matrix products are defined and c be a scalar. Properties of Matrix Multiplication a) A ( BC ) = ( AB ) C b) ( A + B ) C = AC + BC c ) C ( A + B ) = CA + CB d) c ( AB ) = ( c A ) B + A ( c B ) Associativity Matrix Product Right Distributive Property Left Distributive Property

11. LINEAR ALGEBRA Properties of Identity Matrices =   For a positive integer, would denote the square matrix of order n whose main diagonal (left to right) entries are 1 and the rest of the entries are zero ( Identity Matrix ).     =       =  

12. LINEAR ALGEBRA Properties of Identity Matrices a ) A = A   A = =   b ) A = A   A = For a positive integer, and would denote the square matrices of order n and m, respectively, whose main diagonal (left to right) entries are 1 and the rest of the entries are zero ( Identity Matrix ).       =       Multiplicative Identity

13. LINEAR ALGEBRA Let A , B and C be matrices with sizes such that the given matrix products are defined and c be a scalar. Properties of Transposes a) = A   b) = +   c ) =   d) =   Transpose of a transpose Transpose of a sum Transpose of a scalar multiple Transpose of a product

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