MATH 221 Linear Algebra MATH 221 Linear Algebra Spring 2023
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Spring 2023: MATH 221
Linear Algebra
Homework 1
1. (10 pts) Consider the following two matrices:
|
A = l0(1) 「4 |
−2 5 − 1 |
1 −2 3 |
|
−2 1 5 −2 7 − 1 |
8(0)」 −6 |
(a) Write down one way to convert A to B using elementary row operations.
(b) Can we use elementary row operations to convert B into A? If so, how do those operations compare to your answer to part (a)? If not, why not?
2. (25 pts) Consider the augmented matrix:
|
B = l 2(1) 5(2) 「0 c |
0 c 1 |
3(1)」 − 1 |
Find all possible values of c for which the system
(a) has a unique solution.
(b) has no solution.
(c) have infinitely many solutions.
Show all your work and explain your reasoning.
3. (25 pts) For the following homogeneous system
x + 2y + z = 0x + 3y + 6z = 0 2x + 3y + az = 0
find all values of a for which the system has nontrivial solutions. For each value of a write solutions in parametric form and a linear combination of basic solutions. Show all your work and clearly label the steps you take.
4. (40 pts) Determine if each of the following systems has a trivial solution. Try to use as few row operations as possible.
(a)
2x1 − 5x2 +8x3 = 0
−2x1 − 7x2 +x3 = 0
4x1 + 2x2 +7x3 = 0
(b)
−3x1 + 5x2 −7x3 = 0
−6x1 + 7x2 +x3 = 0
(c)
x1 − 3x2 +7x3 = 0
−2x1 + x2 −4x3 = 0
x1 + 2x2 +9x3 = 0
(d)
−5x1 + 7x2 +9x3 = 0
x1 − 2x2 +6x3 = 0
2023-01-29