# 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

## About Matrix Algebra: Manipulating Data Sets and Solving Related Problems

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## Presentation Transcript

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