Matrix Algebra: Manipulating Data Sets and Solving Related Problems
Matrix algebra is a mathematical tool that involves the manipulation of matrices, which are arrays of numbers arranged in rows and columns. This technique is useful for performing operations such as
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About Matrix Algebra: Manipulating Data Sets and Solving Related Problems
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Slide1Matrix Algebra
Slide3MatricesMatrix Addition and Subtraction Matrix Multiplication Identity and Inverse Matrices Determinant of a Square Matrix Applications … and why Matrix algebra provides a powerful technique to manipulate large data sets and solve the related problems that are modeled by the matrices.
Slide5Each element , or entry , a ij , of the matrix uses double subscript notation. The row subscript is the first subscript i , and the column subscript is j . The element a ij is the i th row and the j th column. In general, the order of an m × n matrix is m × n.
Slide10Example:
Slide19Re f e r t o t e x t p g 5 8 3
Slide20An n × n matrix A has an inverse if and only if det A ≠ 0.
Slide22Le t A , B , a n d C b e m a t r i c e s w h o s e o r d e r s a r e s u c h t h a t t h e f o l l o w i n g s u m s , d i f f e r e n c e s , a n d p r o d u c t s a r e d e f i n e d . 1 . C o m m u t a t i v e p r o p e r t y A d d i t i o n : A + B = B + A M u l t i p l i c a t i o n : D o e s n o t h o l d i n g e n e r a l 2 . A s s o c i a t i v e p r o p e r t y A d d i t i o n : ( A + B ) + C = A + ( B + C ) M u l t i p l i c a t i o n : ( A B ) C = A ( B C ) 3 . I d e n t i t y p r o p e r t y A d d i t i o n : A + 0 = A M u l t i p l i c a t i o n : A · I n = I n · A = A
Slide23Le t A , B , a n d C b e m a t r i c e s w h o s e o r d e r s a r e s u c h t h a t t h e f o l l o w i n g s u m s , d i f f e r e n c e s , a n d p r o d u c t s a r e d e f i n e d . 4 . I n v e r s e p r o p e r t y A d d i t i o n : A + ( - A ) = 0 M u l t i p l i c a t i o n : A A - 1 = A - 1 A = I n | A | ≠ 0 5 . D i s t r i b u t i v e p r o p e r t y M u l t i p l i c a t i o n o v e r a d d i t i o n : A ( B + C ) = A B + A C ( A + B ) C = A C + B C M u l t i p l i c a t i o n o v e r s u b t r a c t i o n : A ( B - C ) = A B - A C ( A - B ) C = A C - B C
Slide24Text pg588/589 Exercises#2, 4, 14, 20, 24, and 34
