6. ( t, t, t, t, t)

Transcription

1 3 ( t, 3t, + t, t ) (2t, 1 t, 1+ t, t) 5 o solution ( t, t, t, t, t) 2 CCE UESIS, page a A matrix is an ordered rectangular array of real numbers b A matrix has size (or dimension) m n if it has m rows and n columns c A row matrix is one of size 1 n d A column matrix is one of size m 1 e A square matrix is one of size n n 2 wo matrices are equal if they have the same size and the corresponding entries are equal For example, (1) 1 2 (1 2) (1+ 1) 3 1 A he entries satisfy aij a, that is A is symmetric with respect to the main diagonal ji EXERCISES 2, page he size of A is ; the size of B is 3; the size of C is 1 5, and the size of D is 1 2 a 1 ; a 21 11; a 31 6; a hese are entries of the matrix B he entry b 13 refers to the entry in the first row and third column and is equal to 2 Similarly, b 31 3, and b 3 8 he row matrix is the matrix C he transpose of the matrix C is Systems of inear Equations and atrices

2 C he column matrix is the matrix D he transpose of the matrix D is D [ ] 6 he square matrix is A he transpose is A A is of size 3 2; B is of size 3 2; C and D are of size A is of size 3 2 and C is of size 3 3; therefore, their sum does not exist 9 A+ B 10 2 A 3B C D D 2C atrices 13

3 Systems of inear Equations and atrices

4 x y 2 5 2z ow, by the definition of equality of matrices, u 3 3 u 2 2x 2 3 and 2x 5, or x 5/ 2, y 2 5, and y 7, 2z, and z 2 x 2 z y u ; x2 2+ z 2 y+ 2 ow, by the definition of equality of matrices, x, so x 6 2+ z 2, so z 0 2 2u, so u 1 y+, so y 2 1 x 2 2 3z 10 2y u ; 7 x + 8 2y 15 ow, by the definition of equality of matrices, u 15, so u 15 x+ 8 10, so x 2 2y, so y 2 3z 7, so z 7/3 2 2u 3z 10 u 2 atrices 136

5 2 1 2 y 1 u 3y + 8 2u x 1 2z + 1 ; x12 6z 8 8 ow, by the definition of equality of matrices, 2u, sou 2 x 12 8, so x 20 3y+ 8, so 3y 12, y 6z 8, so 6z 12, and z 2 25 o verify the Commutative aw for matrix addition, let us show that A + B B + A ow, A+ B B A 26 o verify the Associative aw for matrix addition, let us show that A + ( B + C) ( A + B) + C ow, B+ C and A+ ( B+ C) ext, A+ B and ( A+ B) + C ( 3+ 5) A 8A A +5A Systems of inear Equations and atrices

6 28 2 ( A ) 2 ( ) 29 ( A+ B) A+ B 30 2( A 3B) 2 2 A 6B ( 2 ) atrices 138

7 35 r Cross r Jones r Smith a he required matrices are esley esley A, B om om b heir holdings at the end of the year are esley C A+ B om I B (103) A II III We want to find r so that (101 r) A B By the definition of scalar multiplication, we just need to consider the corresponding entries in the first row and first column hus, ( r)(30) r 1 30 and r 5 So the percentage increase is 5% 39 a D A + B C Systems of inear Equations and atrices

8 b E 11 D on 0 a ext Fict Fict Ref Hard A aper on b ext Fict Fict Ref Hard B aper on c ext Fict Fict Ref Hard C aper A A US A W atrices 10

9 3 W B H W A ; W W B B H H H-D Y S K A he sum of all elements in the first row is 100% he sum of all elements in the second row is 100% Harley-Davidson gained the most: or 1% 5 rue Each element in A + B is obtained by adding together the corresponding elements in A and B herefore, the matrix ca ( + B) is obtained by multiplying each element in A + B by c n the other hand, ca is obtained by multiplying each element in A by c and cb is obtained by multiplying each element in B by c and ca + cb is obtained by adding the corresponding elements in ca and cb hus ca ( + B) ca+ cb 6 rue (-1)B is the matrix whose elements are the negatives of the respective elements of B he result now follows by the definition of matrix addition 7 False ake and c 2 hen 1 2 ca and 2 6 ( ca) 8 n the other hand, A ( ca) c rue A is obtained from A by interchanging the rows and columns of A ( A ) is obtained by interchanging the rows and columns of A his leads to the original matrix A hus, ( A ) A 11 2 Systems of inear Equations and atrices

10 USIG ECHGY EXERCISES 2, page atrices 12