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Unformatted text preview: Math 136 Assignment 8 Due: Wednesday, Mar 23rd 1. For each of the following matrices, ﬁnd the inverse, or show that the matrix is not
invertible. 21
MAT[4 4]
1 5 —3
b)B: 1 3‘1
1—4 1
1 2 3 1 2 3
2.LetA= 1 2 4 andB= 6 2 2.
1 1 0 ~110 a) Find det A and det B.
b) Find det(AB). c) Find det(A + B). d) Find A”. 3. Write each of the following matrices and their inverses as a product of elementary
matrices. MM
1 0 0 b)B= 3 10
~1/2 —22 4. Suppose that B is an n X 72 matrix that satisﬁes B5 + B3 + B2 = I. Find an expression
for B"1 in terms of B. 5. Let A and B be n X n matrices. Prove that if (AB) is invertible, then A and B are
invertible. 6. Let A be an n x 72 matrix and {271, . . . ,ﬂn} be a basis for R”. Prove that {A171, . . . ,AUn}
is a basis for IR" if and only if A is invertible. 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 MATLAB, use the C191: command. can be found as follows: OOUTLO
OCDOJ 1
For example, the determinant of the matrix A = 4
7 >> A = [1 2 3; 4 5 6; 7 8 0]
>> 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(10A). (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
11111
111 1111 12222122222
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 72 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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 Math

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