M118 Weeks 9 & 10

M118 Weeks 9 - C HAPTER 6 MATRI X ALGEBRA ALGEBRA A MATRIX is a re ctangular array of num rs be MATRIX 1 1 2 3 1 7 8 3 4 0 6 9 A= 2 3 4 ajk = e in

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Unformatted text preview: C HAPTER 6 MATRI X ALGEBRA ALGEBRA A MATRIX is a re ctangular array of num rs: be MATRIX 1 1 2 3 1 7 8 3 4 0 6 9 A= 2 3 4 ajk = e in the ntry jth row and kth colum n a23 = 9 a41 = 3 0 -1 7 6 A iisa (4 x 3) m s atrix => size= (#rowsx # columns) (#rows A VECTOR is a m atrix with only onerow or onecolum n: VECTOR X= 3 7 0 5 -4 - Row vector Row Y= 9 2 - Column vector MATRIX ADDITION MATRIX 4 7 6 4+2 8 5 0 -2 8+4 0+6 5+2 3-1 0+5 -2-8 0 3 2 4 6 + 0 3 2 -1 5 -8 6 12 7 6 2 = 7+0 6+3 = 7 9 5 -10 SCALAR MULTIPLICATION SCALAR 5 9 0 5 4 4•5 4•9 4•0 4•5 4•4 20 20 36 36 0 20 16 4•6 4•(-1) (-1) 4• 6 3 -1 = 4•3 = 12 -4 24 EG1) Le A = t 4 5 -2 6 0 4 and B = -3 11 -4 -5 6 2 a.) FI ND 3A - 4B b.) Find a m atrix Csuch that 3C+ 2A = B EG2) Le A = t c -1 4 d B= -3 11 -5 6 and C= 9 19 2 0 I f 3A + 2B = C what arethevalue of c and d? , s ROW BY COLUMN MULTIPLICATION ROW 6 8 3 •9 6 7 = 6•9 +8•6 +3•7 = 123 123 NOTE: Thenum r of e s in therow on thele m EQUAL thenum r of e s in thecolum on the be ntrie ft ust be ntrie n right. I N OTHER WORDS …THE NUMBER OF C OLUMNSON THE LEFT MUS EQUAL T THE NUMBER OF ROWSON THE RI GHT!! MATRIX MULTIPLICATION MULTIPLICATION A•B = AB (mx k)•(k x n) = (mx n) For A• B to bede d, the# of colum in A m e fine ns ust qual For thenum r of rows in B! be t he MATRIX MULTIPLICATION MATRIX 3 6 8 0 2 4 • • 4 5 6 9 0 3 = (2 x 3) (3 x 2) = (2 x 2) 1 3 6 8 0 2 4 2 #12 • 5 6 4 9 0 3 = 1 2 #12 = (1st row of A)•(2nd colum of B) n = 3 • 9 + 8 • 0 + 2 • 3 = 33 EG3) A = 5 2 6 6 4 0 B= 1 9 C= 5 2 6 4 3 8 3 -5 0 -4 I f de d, find thefollowing: fine D= 1 0 0 0 1 0 0 0 1 a.) BC b.) C <= NOTE: I n ge ral, B ne c.) ABC AB ≠ BA d.) B2 e DA .) TheI de The ntity Matrix (I ): I (n x n) = (n 1 0 ... 0 0 1 ... 1 ... 0 1 0 0 n rows RULE: A•I = I •A = A n colum ns EG4) A = 5 2 6 6 4 0 B= 1 9 C= 5 2 6 4 3 8 3 -5 0 -4 For e of thefollowing ach to bede d, what are fine t hedim nsions of I ? e a.) C I b.) I B c.) I A EG5) Le t A= 5 2 6 4 B= m 3 9 r and C= 3 9 6 -6 For what value of mand r doe A·B = C? s s EG6) Le t A= 3 5 2 4 X= x y and B = 4 6 For what value of x and y doe A·X = B s s RULE: Any syste of line e m ar quations can bewritte in the n f ormAX = B, whe A is thecoe re fficie m nt atrix for t hesyste , X is theve of variable and B is the m ctor s, ve of constants in thee ctor quations. This formof t hesyste is calle theMatrix Formof thesyste . m d Matrix m EG7) Writethefollowing syste of e m quations in Matrix Form : 3x - 2y + z = 8 2x + 3z = -3 6y - 2z = 0 PROBLEM: GI VEN A S TEM OF EQUATI ONS YS , PROBLEM GI AX = B, HOW C WE S AN OLVE I T FOR X ? S OLUTI ON: REAS BY ANALOGY WI TH ON REAS NORMAL ALGEBRA...THAT I S , NORMAL HOW WOULD YOU S OLVE ax = b FOR x ? EG8) S thefollowing syste of e olve m quations for x and y, 3x + 2y = 4 5x + 4y = 6 … without using the techniques of Chapter 5 ! S TI ON 6.2 EC MATRI X INVERSES INVERSES EG1) S thefollowing syste of e olve m quations for x and y, 3x + 2y = 4 5x + 4y = 6 … without using the techniques of Chapter 5 ! A-1 = t heInverse of A Inverse RULE: A•A-1 = A-1•A = I (A-1 acts likethe“re ciprocal” of (A them n ultiplying ) t he atrix A whe m NOTE: Only n x n m atrice haveinve s (and s rse not all n x n m atrice haveinve s)! s rse How do I find A-1 ? How S TEP 1: S t up theaugm nte m e e d atrix [ A | I ] [LHS| RHS] S TEP 2: Usethele ope gal rations on them atrix le until theLHSis in re d form duce . S TEP 3: I f there of S p 2 is [ I | B ], the sult te n B = A-1. I f not, the A-1 doe e n sn’t xist! EG2) Find A-1 and useit to solvethefollowing syste s m of e quations: A.) x -2z = 2 y +4z = 6 2x - z=7 AX = B => X = A-1• B C 2x - 5y .) =7 x - 3y - z = 5 -x + 2y - z = 4 B.) x -2z = 3 y +4z = 0 2x - z = -1 EG3) Le A = t 3 11 and C= 1 4 2 1 4 -1 Find a 2 x 2 m atrix B such that: a.) AB = C b.) BA = C c.) BC= A ab Give them n atrix A = cd A-1 = ___1___ (ad - bc) d -b -c a EG3) Le A = t 3 11 and C= 1 4 2 1 4 -1 Find a 2 x 2 m atrix B such that: a.) AB = C b.) BA = C c.) BC= A EG4) Le A = t 1 -1 -1 -2 x 2 1and B = -1 0 2 1 1 2 1 0 -1 0 1 For what valueof x is B = A-1 ? EG5) Le A = t 0 1 0 0 1 0 0 0 1 -1 1 0 0 0 1 3 What is thee in these ntry cond row and t hird colum of A-1 ? n ...
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This note was uploaded on 01/24/2011 for the course MATH M118 taught by Professor Stevemckinley during the Fall '07 term at Indiana.

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