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Unformatted text preview: Math 136 Assignment 9 Due: Wednesday, Mar 3lst 1. Calculate the determinant of the following matrices. 0 1 «1 0 1 ——1 5 5 2
3 0 0 2 3 1 2 4 6
a)A= b)B:: —1 «3 8 0 —2
0 1 2 1
5 0 0 7 1 1 2 1 1
1 2 —1 1 0
a b c 1 —1 i
(2)0: a+b 2b c+b d)D:—. 1+z' «2' z'
2 2 2 1—z' —1+2i 1+2i
a+p b+q c+r a b c p q r
2. a) Prove that det d e f : det d e f + det d e f
g h k g h k g h k a + p b + q c + 7"
b) Use part a) to express det d + :r e + y f + 2 as the sum of determinants of
g h k matrices who entries are not sums. 3. n X n matrices A and B are said to be similar if there exists an invertible matrix P
such that P’IAP = B. Prove that if A and B are similar, then det A : det b. 1 3 ——2
4. Determine the inverse of A = 0 1 5 by the cofactor method.
——2 ——6 7
5. Use Cramer’s Rule to solve the following systems.
a) 2$1+$2:1 5$1+IE2—$3:4
3$1+7$22~2 9$1+$2—$3:1
1171 — $2 + 5$3 2 2
1 CL 2 $1 4 1
6. Let A = 0 1 b , f r: :32 and b r: 1 . Assuming that A is invertible,
1 C “1 (I23 2 use Cramer’s Rule to ﬁnd the value of :32 in the solution of the equation A"? : K] . In each case either prove the statement or give an example showing that it is false.
Assume that A and B are n X n matrices. a) det(A + B) = detA + det B.
b) If det A = 0, then rankA < n.
c) det(—A) = —detA d) If A2 = I, then [detAI 2 1. Use MATLAB to complete the following questions. You do not need to submit a printout of your work. Simply use MATLAB to solve the problems,
and submit written answers to the questions along with the rest of your assignment. Determinants To ﬁnd the determinant of a matrix in MATLAB7 use the det command. For example, the determinant of the matrix A = can be found as follows: \Irbr—t
00mm
cacao >>A=[123;456;780]
>>det(A) MATLAB returns that the determinant of A is 27. Question 1 (a) Construct a random 5x5 matrix, A, with integer entries between —9 and 9 (see MATLAB’s
randint command). Compare det(A) with det(AT), det(—A), det(2A), and det(1OA). (b) Repeat part (a) with a, random 6x6 matrix, B , with integer entries. (c) Make conjectures about how these determinants are related. Record your conjectures in
your diary ﬁle. Question 2 (a) Use MATLAB to compute the determinants of the following matrices: 111111
111111111122222 111 12222
1222 123333 122, ,12333,
123 1233 12344 123444
1234 12345 123455
123456 Question 2 continued (b) Using your results from part (a), what is the determinant of the following n x n matrix? 1 1 1 1
12 2 2
12 3 3
12 3 n Record your answer in your diary ﬁle. (C) Confirm your answer in part (b) by using row operations to evaluate the determinant.
Write your solution / explanation including the row operations out on paper and attach it to
your submission. (d) Using your results from the above parts, guess what the determinant of the following matrix is:
1 1 1 1
1 3 3 3
1 3 6 6
1 3 6 . . . 3(n — 1) Record your guess as well as an explanation in your diary ﬁle. ...
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 Spring '08
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 Linear Algebra, Algebra

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