F.6 Mathematics Module 2 - 提供優質補習課 …

1 MathematicsModule 2 Unit 13 Matrices andDeterminants(English Version) SuperMaths Dr. Herbert LamJackie TaiAndrew Yau All Rights HKDSE Mathematics (M2)Headway SuperMathsMatrices and SuperMaths All Rights Reserved1 Unit 13 Matrices and Framework of concepts and algebraic operationsAmatrixAof size (order)m n is a rectangular array of numbers withm(horizontal) rows andn(vertical) columns,111212122212[ ]nnijmmmnaaaaaaAaaaa or 111212122212nnijmmmnaaaaaaaaaa .The elementija(in thethirow and thethjcolumn) is called theij or( , )i jentry of the matricesAandBare said to beequalif they have the same size and thecorresponding entries are of the same sizem n , thenA B , thesumofAandB, is amatrix of sizem n , with its entries equal to the sum of the corresponding entries a matrix of sizem n , and a certain number, thenA is a matrix ofsizem n , with its entries equal to times the corresponding entries isthescalar multipleofAby.

2 By definition,( 1)A BAB . And( 1)B is usually written asB .Let[ ]ijAa be anm n matrix, and[ ]ijBb be ann p thethirow ofAis12[,]iiina aa , andthethjcolumn ofBis12jjnjbbb . HKDSE Mathematics (M2)Headway SuperMathsMatrices and SuperMaths All Rights Reserved2 Thematrix productofAandB, written asAB, is a matrix[ ]ijcof sizem p defined as 11121121222212jpjpnnnjnpbbbbbbbbbbbb =111212122212nnmmmpccccccccc where1 12 2ijijijin njca ba ba b 1ia:1stelement of thethirow :1stelement of thethjcolumn example, taking the2ndrow ofAand the1stcolumn ofB, we obtain2121 1122 2121n nca ba ba b .Note:The definition of matrix multiplication requires that the number ofcolumnsof the first factorAbe the same as the number of rows ofthe second factorBin order to form the productAB.

3 If thiscondition is not satisfied, the product is HKDSE Mathematics (M2)Headway SuperMathsMatrices and SuperMaths All Rights Reserved3 Example 1 Given the matrices,112034251A ,01 110 21 2 1B ,112C ,compute the followingA B ,A C ,3BA ,AB,AC, :1 01 12 110 30 13 04 213 62 1 5 21 13 7 0A B .SinceAis3 3 ,Cis3 1 , thusA C is 1336310 209121 2 16 153BA 0 31 ( 3)1 63451 00 92 1219101 ( 6)2 151 ( 3)5134 .1 0 ( 1) 1 2 ( 1)1 1 ( 1) 0 2 21 1 ( 1) 2 2 10 0 3 1 4 ( 1)0 1 3 0 4 20 1 3 2 4 1( 2) 0 5 1 ( 1) ( 1) ( 2) 1 5 0 ( 1) 2 ( 2) 1 5 2 ( 1) 1AB 3511810647.

4 1 1 ( 1) ( 1) 2 260 1 3 ( 1) 4 25( 2) 1 5 ( 1) ( 1) 29AC .Cis3 1 ,Ais3 3 , and1 3 , henceCAis HKDSE Mathematics (M2)Headway SuperMathsMatrices and SuperMaths All Rights Reserved4 Special MatricesRow Matrix: is a matrix with only 1 row, a1n matrix. 2 5, 3 4 Matrix:is a matrix with only 1 column, a1n matrix. , 568 .Zero Matrix (or null matrix): is a matrix with all its entries equal ,0 00 0 The zero matrix of orderm n is denoted byor simply .m n 00 Square Matrix:is a matrix with equal number of rows and columns. An n squarematrix is said to be of 24 7 ,0 257 192 43 are square matrices of order 2 and 3 Matrix:is a square matrixAwith all its entriesija, whereij , equal 0004 Identity Matrix or Unit Matrix:is a diagonal matrixAwith all its entries1iia.

5 Aunit matrix of ordernis denoted byInor 0 01 0,0 1 00 10 0 1II HKDSE Mathematics (M2)Headway SuperMathsMatrices and SuperMaths All Rights Reserved5 General properties of matricesLetA,B, andCbe matrices, and ,1 &2 be scalars. Provided the operationsmake sense, BBA (Commutative law of addition)2.()()AB CA BC (Associative law of addition) a zero matrix, then0 0 AAA (Additive identity) any matrixA,( 1)()0 AAAA , a zero matrix(Additive inverse) 2()( )AA ()AAA 7.()A BAB 8.() ()A BCAB C (Associative law of multiplication)*9.()A B CAC BC (Distributive law: Multiplication is right distributive over addition )*10.()A B CABAC (Distributive law: Multiplication is left distributive over addition )11.()()A BAB 12.

6 ()()A BAB 13. LetIbe an identity matrix. ThenAIA andIAA .*14. The transpose of a matrix, 213242TA ,13B 1 3TB .*15.()TTTABB A HKDSE Mathematics (M2)Headway SuperMathsMatrices and SuperMaths All Rights Reserved6 Example 2 Findna bb a .SolutionLet0 11 0P . Obviously2PI . Alsoa baI bPb a .Then by the Binomial theorem,12 223 33123()nnnnnnnnnnna baI bPa I C a bP C ab PC a b Pb Pb a 2 24 413 32413()()nnnnnnnnnaC abC abIC a b C abP 12 212 212121() ()2nnnnnnnnnnaC a b C abaC a b C abI 12 212 212121() ()2nnnnnnnnnnaC a b C abaC a b C abP ()()()()22nnnna ba ba ba bIP ()()()()12()()()()nnnnnnnna ba ba ba ba ba ba ba b.

7 Note:Another method of findingna bb a is to make use of a matrix1111P and its inverse1P . It is given in Example 5, Section Cof this HKDSE Mathematics (M2)Headway SuperMathsMatrices and SuperMaths All Rights Reserved7 Instant Drill the following expressions:a) 253412 b) 510240623c) 1758242264133d) 104302001P, 180022101 Qand 421212124R. Evaluate each of the followinga)RQP543 b)RQPRQPR32243 c)PRQRQP HKDSE Mathematics (M2)Headway SuperMathsMatrices and SuperMaths All Rights Reserved8 Exercise is given that 1352 Aand 312dcbaB. IfBA , find the values ofa,b, is given that yzxA404and cbaccbaaB232. IfBA 2, find the values ofa,b,c,x, 1350311402kt. Find the values 243632480001230ts. Find the values yyx8111527423. Find the values 023zyxA. If 42714 Aand 1331A, HKDSE Mathematics (M2)Headway SuperMathsMatrices and SuperMaths All Rights ConceptsLet11122122aaAaa.

8 Then thedeterminantofA, denoted byA,det( )A, or11122122aaaaequals, by definition,11 2212 21a aa a .The determinant of an3 3 matrix is defined in terms of determinants of2 2 matrices. Thus let111213212223313233aaaAaaaaaa , then the determinant ofAis given by222321232122111213323331333132aaaaaaaa aaaaaaa .The formula for defining the determinant of a larger size square matrix, in terms ofsmaller size matrices, is given as follows11121121222112( 1)( )( 1)( )nni jijijjnni jijijinnnnaaaaMiaaaaaMjbaaa for anyfor anywhereijMdenotes the determinant of the submatrix obtained from a square matrix()ijn nAa by deleting thethirow and thethjcolumn ofA(yielding a squarematrix of order1n ).( 1)i jijijCM is called thecofactorofija,andijMis called theminorofija.(a) & (b) are called, respectively, the cofactor (minor) expansion ofdet( )Aby thethirow andthjcolumn.

9 It follows that(i)det()det( )TAA (ii)det( )0A ifAhas a zero row or zero proof is quite simple and is left to the HKDSE Mathematics (M2)Headway SuperMathsMatrices and SuperMaths All Rights Reserved10 Example 3 Let12 31 3 113 2A . Find the minors and cofactors ofAand :113 133 2M ,121 1312M ,131 3613M ,212 353 2M ,221 311 2M ,231 211 3M 312 373 1M ,321341 1M ,331251 3M .Hence113C ,123C ,136C ,215C ,221C ,231C ,317C ,324C ,335C .We can use cofactor expansion by any row or column, 1111131223Aa Ca Ca C (expansion by the1strow)(1)(3) (2)(3) (3)( 6)9 ,or112232322222a Ca Ca C (expansion by the2ndcolumn)(2)(3) (3)( 1) (3)( 4)9 .Note:To evaluate a3 3 determinant, we can use theRule of 22 3312 23 3113 21 3231 22 1332 23 1133 21 12det( )Aa a aa a aa a aa a aa a aa a a.

10 HKDSE Mathematics (M2)Headway SuperMathsMatrices and SuperMaths All Rights Reserved11 Instant Drill Consider210513142.(a) Find23M.(b) Find31A.(c) Find the value of each of the following 3. Find the value of each of the following determinants by cofactor HKDSE Mathematics (M2)Headway SuperMathsMatrices and SuperMaths All Rights properties of determinantsAside from using cofactor expansion to evaluate determinants, the followingproperties are often helpful in LetAbe of the form11121222000nnnnaaaaaa , entries below the diagonal are 22detnnAAa aa .2. LetAbe ann n matrix. AndBis obtained by interchanging any two rows(or columns) ofA. ThenAB . one row ofAis a multiple of another row, then0A .4. IfBis the matrix obtained by adding a multiple of one row (or column) ofAto another row (column), thenBA.