Operations with Matrices


 Allen Harvey
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1 Section. Equlit of Mtrices Opertions with Mtrices There re three ws to represent mtri.. A mtri cn be denoted b n uppercse letter, such s A, B, or C.. A mtri cn be denoted b representtive element enclosed in brckets, such s [ ij ], [b ij ], or [c ij ].. A mtri cn be denoted b rectngulr rr of numbers such s A = [ ij ] = m m m n n n mn Definition: Two mtrices A = [ ij ] nd B = [b ij ] re equl if the hve the sme order (m n) nd [ ij ] = [b ij ] for ll i =,,, m nd j =,,, n. In other words, if ll of the corresponding entries re equl.. but
2 Testing Mtri Equlit using Grphing Clcultor. Press [ nd ] [MATRX] [A]. Press [ nd ] [TEST] [=]. Press [ nd ] [MATRX] [B] [ENTER] If the mtrices re of the sme order, nd ll corresponding elements re equl, then the clcultor will return the vlue. CHAT PreClculus Section. If the mtrices re of the sme order, nd ll corresponding elements re not ll equl, then the clcultor will return the vlue. If the mtrices do not hve the sme dimensions, ou will get nd error messge tht ss DIM MISMATCH Coping Mtri To plce the contents of mtri A into the mtri B, do the following:. Press [ nd ] [MATRX] [A]. Press [STO]. Press [ nd ] [MATRX] [B] [ENTER] A nd B re now identicl mtrices.
3 Section. Mtri Addition Definition of Mtri Addition: If A = [ ij ] nd B = [b ij ] re mtrices of order m n, their sum is the m n mtri given b A + B = [ ij + b ij ]. *The sum of two mtrices of different orders is undefined. Emple: Find the following sums. ) b) Solutions:
4 Section. Sclr Multipliction In opertions with mtrices, numbers re usull referred to s sclrs. Definition of Sclr Multipliction If A = [ ij ] is n m n mtri nd c is sclr, the sclr multiple of A b c is the m n mtri given b ca = [c ij ] Definition: The smbol A represents the dditive inverse of A nd equls ()A. Moreover, A B = A + (B). Emple: Consider the mtrices: A nd B
5 Section. Find the following: ) A A b) B ) ( ) ( B B c) A B B A
6 Section. Mtri Opertions on the Grphing Clcultor Adding Mtrices If the mtrices hve the sme dimensions, the cn be dded. To dd mtri A nd mtri B, do the following:. Press [ nd ] [MATRX] [A]. Press [+]. Press [ nd ] [MATRX] [B] [ENTER] The resulting mtri is A + B. ***The sme method is used for subtrcting mtrices. Note: If the mtrices do not hve the sme dimensions, ou will get nd error messge tht ss DIM MISMATCH Sclr Multipliction To multipl sclr times the mtri A, do the following:. Enter the sclr vlue. Press [*] [ nd ] [MATRX] [A] [ENTER] Note: You cn multipl the sclr before or fter the mtri.
7 Section. Negting Mtri To chnge the signs on ll of the elements of mtri, do the following:. Enter the negtion smbol [()]. Press [ nd ] [MATRX] [A] [ENTER] Emple: Enter the following mtrices on our clcultor: A nd B Find the following: ) A + B b) B A B B c) A d) A  B A A B
8 Section. Definition: The ero mtri is the m n mtri given b O = []. The ero mtri cn be n sie, nd consists entirel of eros. The ero mtri is lso the dditive identit for the set of ll m n mtrices. Properties of Mtri Addition nd Sclr Multipliction Let A, B, nd C be m n mtrices nd let c nd d be sclrs.. A + O = O + A = A. A + B = B + A. A + (B + C) = (A + B) + C. (cd)a = c(da). A = A. c(a + B) = ca + cb. (c + d)a = ca + da Note: # nd # lso men tht we cn fctor out common fctor for n mtri. The lgebr of rel numbers nd the lgebr of mtrices hve mn similrities. Emple: Solve the mtri eqution A X = B, where A nd B
9 Section. Solving for X we get X = (B A). Emple: Solve B X = A using the mtrices A nd B s given bove. A B X X
10 Section. Mtri Multipliction Definition of Mtri Multipliction: If A = [ ij ] is n m n mtri nd B = [b ij ] is n n p mtri, the product AB is n m p mtri given b AB = [c ij ] where c ij = i b i + i b i + i b i in b nj. Wht we re doing is tking the elements in the ith row of A, multipling them b the corresponding elements in the jth column of B, nd then summing these products.
11 Section. Emple: Multipl For our product, we will multipl ech row of the first mtri times ech column of the nd mtri. ( )( ) ()( ) ()( ) ( )( ) ()( ) ()( ) ( )() ()() ()() ( )() ()() ()() This entr in row, column cme from multipling row of the first mtri times column of the nd mtri. Note: In order for the product to be defined, the number of columns of the first mtri must equl the number of rows of the nd mtri. A B = AB m n n p m p Equl Dimensions of AB
12 Section. Emple: Find the following products. ) b) c) d) = undefined e)
13 Section. f) g) *Notice tht for f) nd g) tht we did not get the sme nswer. Even if AB nd BA re defined, mtri multipliction is, in generl, not commuttive. Mtri Multipliction on the Grphing Clcultor If the number of columns of the first mtri equls the number of rows of the nd mtri, the mtrices cn be multiplied. (An error messge will result otherwise.). Press [ nd ] [MATRX] [A]. Press [*]. Press [ nd ] [MATRX] [B] [ENTER] The resulting mtri is A B.
14 Section. Emple: Enter the following mtrices on our clcultor: A nd B Find AB. AB Definition: The identit mtri of order n is n n n mtri tht consists of s on its min digonl nd s everwhere else. It is denoted I n. If it is lred understood tht the mtri is squre, we cn refer to it s simpl I. emples:,, nd
15 Section. Emple: Multipl ()() ()() ()() ()() ()() ()() ()() ()() Multipling b the identit mtri gives bck the mtri we strted with. Properties of Mtri Multipliction Let A, B, nd C be mtrices nd let c be sclr.. AI n = I n A = A. A(BC) = (AB)C. A(B + C) = AB + AC. (A + B)C = AC + BC. c(ab) = (ca)b = A(cB)
16 Section. Applictions Notice how the sstem Cn be written s the mtri eqution AX = B, where A is the coefficient mtri nd X nd B re column mtrices. =
17 Section. Emple: Write the following sstem of equtions s mtri eqution AX = B. Solution: The mtri eqution is: Emple: Write the following sstem of equtions s mtri eqution AX = B. Then use Guss Jordon elimintion on the ugmented mtri [A:B] to solve for the mtri X. Solution:
18 Section. The ugmented mtri [A:B] is: or Solving gives: The solution is X = =
19 Section. Emple: Two tennis tems submit equipment requests to their sponsors. Women s Tem Men s Tem Blls Rckets Shoes Ech cn of blls costs $, ech rcket costs $, nd ech pir of shoes costs $. Use mtri multipliction to find the totl cost of equipment for ech tem. Solution: So, the totl cost of equipment for the women s tem is $, nd the totl cost of equipment for the men s tem is $,.
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