Chapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY

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1 Chpter 3 MTRIX In this chpter: Definition nd terms Specil Mtrices Mtrix Opertion: Trnspose, Equlity, Sum, Difference, Sclr Multipliction, Mtrix Multipliction, Determinnt, Inverse ppliction of Mtrix in Solving Liner Eqution System Determinnt of 3 x 3 Mtrices 3. MTRIX NOTTION ND TERMINOLOGY Defnition. mtrix is rectngulr rry of numbers. The numbers in the rry re clled the entries in the mtrix. Exmples: Nottion Generlly written s: m n= ij = C = D = mtrix with 2 rows, nd 3 columns NOT mtrix mtrix with 3 rows, nd 3 columns 2 3 D 3 3

2 n rows nd n columns n n, is squre mtrix. SPECIL MTRICES The following re severl specil mtrices. Generl form Nme Properties Exmples 6 7 nd = n [ n] Row mtrix Only one row [ ] 2 [ ] 2 n = n Column mtrix Only one column 3 nd m n= Null mtrix ll entries re null , D I n d d dn 2 n = = 0 0 Digonl mtrix Identity mtrix squre mtrix in which ll of the entries off the min digonl re zero, i.e. = 0, i j ij squre mtrix, with ll digonl entries re, nd zero elsewhere , nd I I = 0, 0 0 =

3 n n n 22 2n = nn Upper tringulr mtrix squre mtrix in which ll the entries below the min digonl re zero , n n n n n n2 nn 2 22 = 2 n n = n n2... nn Lower tringulr mtrix Symmetric mtrix squre mtrix in which ll the entries bove the min digonl re zero squre mtrix in which ij = for ll i nd j. ji , OPERTIONS ON MTRICES Definition. Two mtrices re defined to be equl if they hve the sme size nd their corresponding entries re equl. In mtrix nottion, if only if ( ) ( B) ij ij = ij nd B bij =, or equivlently ij = bij for ll i nd j. = hve the sme size, then = B if nd Exmple: Consider the mtrices 3 3 = 2 x, B 3 4 = 2 7, nd C = Is = B? Why? 3. Is = C? Why? 2. Is B = C? Why? 3

4 Definition. If then: = ij nd B bij = re mtrices of the sme size, ( + B) = ( ) + ( B) = ij + bij, nd ( B) = ( ) ( B) = ij bij ij ij ij ij ij ij Exmple: Consider the mtrices 4 0 = 0 5, 9 7 B = 4 5 nd 3 9 C = 6 2 Then: ( 7) B= + = = , B=... Wht bout: + C? C B? Definition. If = is ny mtrix nd c is ny sclr, then the product c is the ij mtrix obtined by multiplying ech entry of by c. In mtrix nottion, if = ij then ( c) c( ) c ij ij ij = =. Exmple. For the mtrices We hve = 9 2 0,nd B = 4 5 2(3) 2( 8) 2(6) = = 2(9) 2(2) 2(0) , =

5 Wht bout: 5 = 3 = B = 2 B = 2+ 3B= + 2B = Definition. If = is n m x r mtrix nd B is n r x n mtrix, then the product ij B is the m x n mtrix whose entries re determined s follows. B = sum of multipliction of corresponding entries in row i of nd Entry of ( ) ij column j of B. The definition of mtrix multipliction requires tht the number of columns of the first fctor be the sme s the number of rows of the second fctor B in order to form the product B. If this condition is not stisfied, the produt is undefined. Exmple: For the mtrices = Let C = B. nd B b b b b b b = b3 b32 b33 b b b , 5

6 Then b b2 b b2 b22 b 23 C = B = c c2 c = 24 b3 b32 b33 c2 c22 c 23 b4 b42 b43 nd The entry of row 2 nd column 3 of C is: c23 = 2b3 + 22b b b43 Cn B be clculted? Exmple. Consider the mtrices = 3 7, nd B = Then: B = 2 8 4( 6) + ( 2)(2) + 5(3) 4(4) + ( 2)(8) + 5(9) 3 7 = 3( 6) (2) 7(3) 3(4) (8) 7(9) = = B = TRNSPOSE OF MTRIX T Definition. If is ny m x n mtrix, then the trnspose of, denoted by, is defined to be the n x m mtrix tht results from interchnging the rows nd the columns of. In mtrix nottion, ( T ) ( ) ij =. ji 6

7 Exmple. Consider the mtrices = 4 8 nd B = Then: T 0 4 = T B =... Exmine: Is T T =? Why? Is B = B? Why? TRCE OF SQURE MTRIX Definition. If is squre mtrix, then the trce of, denoted by tr( ), is defined to be the sum of the entries on the min digonl of. The trce of is undefined if is not squre mtrix. Exmple. The following re exmples of mtrices nd their trces = 4 0 5, then tr ( ) = = B = 5 83, then tr ( B ) = = 70 7

8 EXERCISE. Suppose tht, B, C, D, nd E re mtrices with the following sizes: 4 5 B4 5 C5 2 D4 2 E. 5 4 Determine which of the following mtrix expressions re defined. For those which re defined, give the size of the resulting mtrix.. B e. E(+B) b. C + D f. E(C) c. E + B g. E T d. B + B h. ( T +E)D 2. Solve the following mtrix eqution for, b, c, nd d. b b+ c 8 = 3d + c 2 4d Consider the mtrices =, B = 0 2, C = 3 5, D = 0, E = Compute the following (where possible).. D + E b. 2B-C 8

9 c. -3(D+2E) g. (3E)D d. 4tr(7B) h. B e. 2 T +C i. B f. 2E T 3D T j. (2D T -E) 4. Find the 3 x 3 mtrix = ij whose entries stisfy the stted condition., if i j >. ij = i+ j b. ij =, if i j 5. Given tht 3 + b b is symmetric mtrix, find nd b

10 DETERMINNT OF 2 X 2 MTRIX Definition. Let b = c d, then the determinnt of is d bc, tht is det() = d bc In mtrix nottion, the determinnt of cn be written s, b det() = = = d bc c d Exmple: Consider the following mtrices: = 4 b. B = 4 3 Then: c. 3 6 C = 2. det() = 4 3() 9(4) = = = b. det(b) =... c. det(c) =... Exmple: Find x so tht the determinnt of x 2 4 = x is 8. Solution: x 2 4 Det() = = ( x 2)( 3) 4( x+ 3) = 3x+ 6 4x 2= 7x 6 x Since the determinnt is 8, then 7x 6= 8 7x = 4 x = 2 0

11 . Find x so tht the determinnt of x 2 4 = x is Find x so tht the determinnt of x + 2 B = 3 x 5 is EXERCISE. Evlute the determinnt d b e c f Find x so tht the determinnt of 3 x 4 is 8. x 2

12 3. Find ll possible vlues for so tht the determinnt of is Consider the mtrices 2 4 =, B =. Evlute: 4 3. det() d. det() + det(b) b. det(b) e. det(b) c. det( + B) f. det()det(b) INVERSE OF 2 x 2 MTRIX Definition. If is squre mtrix, nd if mtrix B of the sme size cn be found such tht B = B = I, then is sid to be invertible nd B is clled n inverse of. In mtrix nottion, the inverse of is, so if B = B = I, then = B Hence, = = I 2

13 Exmple: The mtrix = 2 is n inverse of B = 3, since nd B = I 2 = = = B= = = = I Theorem. The mtrix b = c d is invertible if d bc 0, in which cse the inverse is given by the formul d b = d bc c non-invertible mtrix, or mtrix with the determinnt = 0 is clled singulr mtrix. Wht is non-singulr mtrix? Exmple. Consider the following mtrices: 3 5 = 2 B 6 4 = C = 6 9 Then the inverses re: = =. = B =... C =... 3

14 Properties of n inverse If nd B re non-singulr squre mtrices of the sme size, then ( B) = B Exmple: 2 Let = 2 3 nd Then, B 2 0 = 3 On the other hnd, B = 2 3 = nd 3 2 ( B) = 2 ( 0) = = = = 3 ( 4) B 2 nd B = = = = 3 2 = Both give the sme result, which verify tht ( B) = B EXERCISE. Determine whether is the inverse of B, or vise vers =, B 4 3 = b. =, B = 3 5 4

15 2. Evlute whether the following mtrices re singulr or non-singulr. If it is non-singulr mtrix, find the inverse d b. 5 4 e Let 3 4 =, B =. Find: 4 0. nd B c. (B) b. B d. B 5

16 Let =. Find: 2 4. T c. ( T ) b. d. ( ) T 5. Given the following informtion, find. 9. = 4 b. 6 7 (3 ) = 7 8 6