Name: Linear Algebra – 3191 Summer 2012 – Sullivan Homework ...

Name:
Linear Algebra – 3191
Summer 2012 – Sullivan
Homework # 3
Due Date: June 26, 2012
HIGHLY SUGGESTED PRACTICE PROBLEMS:
• section 2.3: 1-9odds, 11, 13, 17, 25
• section 2.5: 1, 5, 9, 21, 25, 31
• section 3.1: 3, 11, 13, 15, 31, 32, 37, 39, 41
• section 3.2: 1, 3, 5, 11, 15, 17, 19, 21, 25, 29, 31, 32
Solutions to suggested True/False:
#2.3.11: T,T,F,T,T; #3.1.39: T,F; #3.2.27...T,T,T,F;
ASSIGNED PROBLEMS:
1. Determine if the matrix is invertible. Use as few calculations as possible and be sure to explain your reasoning.
(Soln) ¡+your solution here+¿
2. Consider the matrix
⎛ ⎞
1 −3 −6
⎜ ⎟
⎝ 0 4 3 ⎠
−3 6 0
⎛ ⎞
−5 0 4
⎜ ⎟
A = ⎝10
2 −5⎠
10 10 16
(a) Find an LU factorization of A. Use Wolfram Alpha to check your answer. (a solution with no work will
receive no credit)
(b) Use the LU decomposition from part (a) to solve the equation Ax = b where
(Soln) (a) ¡+your solution here+¿
(b) ¡+your solution here+¿
⎛ ⎞
14
⎜ ⎟
b = ⎝−2⎠
3. Read sections 3.1 and 3.2 and answer the following equations about the determinant:
Let A be a square matrix.
(a) det(I) =??
(b) If a multiple of one row of A is added to another row to produce matrix B then det(B) =??
156
(c) If two rows of A are interchanged to produce B then det(B) =??
1

(d) If one row of A is multiplied by k to produce B, then det(B) =??
(e) If matrix A has a row of zeros then det(A) =??
(f) If matrix A has two equal rows then det(A) =??
(g) The determinant of a triangular matrix is
(h) det(AB) =??
(i) det(A T ) =??
(j) A is invertible if and only if
4. Use cofactor expansion (described in section 3.1) to compute the determinant
(Soln) ¡+your solution here+¿
5
−2 4
0
3 −5
2
−4 7
5. Find a formula for det(r·A) when A is an n×n matrix and r ∈ R. Explain your reasoning.
(Soln) ¡+your solution here+¿
2