#61 | sin cos1:40Done elementary_linear_algebra_8th_...50Chapter 2 Matrices61. Guided Proof Prove that if A and B are diagonal72. Show that no 2 x 2 matrices A and R exist that samatrices (of the same size), then AR = RA.the matrix equationAB - BA =Getting Started: To prove that the matrices AB and BAare equal, you need to show that their correspondingentries are equal.(i) Begin your proof by letting A = [a,] and B = [b,]be two diagonal n x n matrices.(ii) The ijth entry of the product AB is73. Exploration Let i = /-1 and let-: and B=(a) Find A. A', and A*. (Note: A? =A' = AAA - A'A, and so on.) Identify any similamwith f, P, and i.(iii) Evaluate the entries cy for the two cases i + j andi = j.(b) Find and identify B.74. Guided Proof Prove that if the product AB is a sqmatrix, then the product RA is defined.Getting Started: To prove that the product Bdefined, you need to show that the number of coluof R equals the number of rows of A.(iv) Repeat this analysis for the product BA.62. Writing Let A and R be 3 x 3 matrices, where A isdiagonal.(a) Describe the product AB. Illustrate your answerwith examples.(i) Begin your proof by noting that the numbecolumns of A equals the number of rows of R.(b) Describe the product BA. Illustrate your answerwith examples.(ii) Then assume that A has size m x n and B has(c) How do the results in parts (a) and (b) change whenthe diagonal entries of A are all equal?nx p.Use the hypothesis that the product ABsquare matrix.Trace of a Matrix In Exercises 63-66, find the75. Proof Prove that if both products AB and BAdefined, then AR and RA are square matrices.trace of the matrix. The trace of an n xn matrix Ais the sum of the main diagonal entries. That is,Tr(A) = a, + an + + a[163. |о -2[3[176. Let A and B be matrices such that the product Adefined. Show that if A has two identical rows, then[164. 023corresponding two rows of AB are also identical.477. Let A and R be nxn matrices. Show that if therow of A has all zero entries, then the ith row ofwill have all zero entries. Give an example using 23[142432-120 6matrices to show that the converse is not true.65.46.36 20 5[21 1 -378. CAPSTONE Let matrices A and R besizes 3 x 2 and 2 x 2, respectively. Answer eaquestion and explain your answers.67. Proof Prove that each statement is true when A and Bare square matrices of order n and c is a scalar.(a) Tr(A + B) = Tr(A) + Tr(B)(a) Is it possible that A = B?(b) Is A + R defined?(b) Tr(cA) = cTr(A)(c) Is AB defined? If so, is it possible that AB = BA68. Proof Prove that if A and B are square matrices oforder n, then Tr(AB)- Trí(BA).69. Find conditions on w, x, y, and z such that AB = BA for79. Agriculture A fruit grower raises two crops, apand peaches. The grower ships each of these cropthree different outlets. In the matrix125 100 75]the matrices belowand B =100 175 12570. Verify AB = BA for the matrices below.a, represents the number of units of crop i thatgrower ships to outlet j. The matrixB = [53.50 $6.00]cos a -sin acos B - sin BA =and B=sin acos asin Bcos B]71. Show that the matrix equation has no solution.represents the profit per unit. Find the product RAstate what each entry of the matrix represents.Cde 27 Conr kemine AI Re Reened Mehe id d rd in wepn Derlec e, w pe me he ed e hEd a ded s ly ling pi Congae Leng ene y2.1 Exercise80. Manufacturing A corporation has three factories,each of which manufactures acoustic guitars andelectric guitars. In the matrixBlock Multiplication In Exercises 83 and 84.block multiplication of matrices A and R.A and B are each partitioned into four submat[Bu B12-B.. B.A =| 70 50 2570135 100A = [A Au and B=A..A..

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50 Chapter 2 Matrices 61. Guided Proof Prove that if A and B are diagonal matrices (of the same size), then AR - RA Getting Started: To prove that the matrices AB and BA are equal, you need to show that their corresponding entries are equal. () Begin your proof by letting A = [a,) and B = [b,] be two diagonal nxn matrices. (ii) The tth entry of the product AB is (iii) Evaluate the entries e, for the two cases i j and i-j. (iv) Repeat this analysis for the product BA.

50 Chapter 2 Matrices 61. Guided Proof Prove that if A and B are diagonal matrices (of the same size), then AR - RA Getting Started: To prove that the matrices AB and BA are equal, you need to show that their corresponding entries are equal. () Begin your proof by letting A = [a,) and B = [b,] be two diagonal nxn matrices. (ii) The tth entry of the product AB is (iii) Evaluate the entries e, for the two cases i j and i-j. (iv) Repeat this analysis for the product BA.

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Chapter2: Second-order Linear Odes

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#61

| sin cos
1:40
Done elementary_linear_algebra_8th_...
50
Chapter 2 Matrices
61. Guided Proof Prove that if A and B are diagonal
72. Show that no 2 x 2 matrices A and R exist that sa
matrices (of the same size), then AR = RA.
the matrix equation
AB - BA =
Getting Started: To prove that the matrices AB and BA
are equal, you need to show that their corresponding
entries are equal.
(i) Begin your proof by letting A = [a,] and B = [b,]
be two diagonal n x n matrices.
(ii) The ijth entry of the product AB is
73. Exploration Let i = /-1 and let
-: and B=
(a) Find A. A', and A*. (Note: A? =
A' = AAA - A'A, and so on.) Identify any similam
with f, P, and i.
(iii) Evaluate the entries cy for the two cases i + j and
i = j.
(b) Find and identify B.
74. Guided Proof Prove that if the product AB is a sq
matrix, then the product RA is defined.
Getting Started: To prove that the product B
defined, you need to show that the number of colu
of R equals the number of rows of A.
(iv) Repeat this analysis for the product BA.
62. Writing Let A and R be 3 x 3 matrices, where A is
diagonal.
(a) Describe the product AB. Illustrate your answer
with examples.
(i) Begin your proof by noting that the numbe
columns of A equals the number of rows of R.
(b) Describe the product BA. Illustrate your answer
with examples.
(ii) Then assume that A has size m x n and B has
(c) How do the results in parts (a) and (b) change when
the diagonal entries of A are all equal?
nx p.
Use the hypothesis that the product AB
square matrix.
Trace of a Matrix In Exercises 63-66, find the
75. Proof Prove that if both products AB and BA
defined, then AR and RA are square matrices.
trace of the matrix. The trace of an n xn matrix A
is the sum of the main diagonal entries. That is,
Tr(A) = a, + an + + a
[1
63. |о -2
[3
[1
76. Let A and B be matrices such that the product A
defined. Show that if A has two identical rows, then
[1
64. 0
2
3
corresponding two rows of AB are also identical.
4
77. Let A and R be nxn matrices. Show that if the
row of A has all zero entries, then the ith row of
will have all zero entries. Give an example using 2
3
[1
4
2
4
3
2
-1
2
0 6
matrices to show that the converse is not true.
65.
4
6.
3
6 2
0 5
[2
1 1 -3
78. CAPSTONE Let matrices A and R be
sizes 3 x 2 and 2 x 2, respectively. Answer ea
question and explain your answers.
67. Proof Prove that each statement is true when A and B
are square matrices of order n and c is a scalar.
(a) Tr(A + B) = Tr(A) + Tr(B)
(a) Is it possible that A = B?
(b) Is A + R defined?
(b) Tr(cA) = cTr(A)
(c) Is AB defined? If so, is it possible that AB = BA
68. Proof Prove that if A and B are square matrices of
order n, then Tr(AB)- Trí(BA).
69. Find conditions on w, x, y, and z such that AB = BA for
79. Agriculture A fruit grower raises two crops, ap
and peaches. The grower ships each of these crop
three different outlets. In the matrix
125 100 75]
the matrices below
and B =
100 175 125
70. Verify AB = BA for the matrices below.
a, represents the number of units of crop i that
grower ships to outlet j. The matrix
B = [53.50 $6.00]
cos a -sin a
cos B - sin B
A =
and B=
sin a
cos a
sin B
cos B]
71. Show that the matrix equation has no solution.
represents the profit per unit. Find the product RA
state what each entry of the matrix represents.
Cde 27 Conr kemine AI Re Reened Mehe id d rd in wepn Derlec e, w pe me he ed e h
Ed a ded s ly ling pi Congae Leng ene y
2.1 Exercise
80. Manufacturing A corporation has three factories,
each of which manufactures acoustic guitars and
electric guitars. In the matrix
Block Multiplication In Exercises 83 and 84.
block multiplication of matrices A and R.
A and B are each partitioned into four submat
[Bu B12
-B.. B.
A =| 70 50 25
701
35 100
A = [A Au and B=
A..
A..

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