Linear Algebra Questions

Math 2568 Autumn 2020
Homework 2
Problem 1 Let A =

2 1
3 4
. Find all matrices B =

a b
c d
satisfying the equation A ∗ B = B ∗ A as
follows:
i) Use the equality A ∗ B = B ∗ A to write down a 4 × 4 system of equations in the unknowns a, b, c, d.
ii) Solve the system of equations in i) using whatever method you prefer. Express your answer as a
parametrized set of solutions, indicating explicitly what variables are being used for the parametrization.
Problem 2 Let A =

1 3
2 7
.
i) Apply row operations to [A | I] to find the inverse of A.
ii) Use the sequence of row operations to express A−1 as a product of elementary matrices, using the
notation for elementary matrices in the textbook.
iii) Using ii), express A explicitly as a product of elementary matrices (again using the textbook notation).
Problem 3 Let A =

(a − 4) −1
2 (a − 1)
.
i) Use row operations to put A into upper triangular form using only type I and III operations [upper
triangular means that A(i, j) = 0 whenever i > j. Make sure your operations are defined for all possible
values of a (suggestion: avoid dividing by any expression which could be zero for certain values of a).
ii) Use i) to find the values of a for which the matrix A is singular.
iii) Assuming a to be a value different from those listed in ii), find an explicit expression for A−1
in terms
of the indeterminate a, by computing rref([AI]).
Problem 4 A system of equations is given by
18×1 + 18×2 − 10×3 + 7×4 − 11×5 + 50×6 = 26
−10×1 − 10×2 + 6×3 − 4×4 − x5 − 27×6 = −13
7×1 + 7×2 − 4×3 + 5×4 + 2×5 + 30×6 = 18
2×1 + 2×2 − x3 + 2×4 + x5 + 12×6 = 8
i) Represent the system as a matrix equation of the form A∗x = b. Indicate precisely what the coefficient
matrix A is, as well as the vector b.
ii) Compute rref([A b]) and use this to write down the general solution of the system in vector form.
Problem 5 Consider the matrix equation A ∗ x = b where A =


14 −8 3 −49 29
−8 5 −2 29 −16
3 −2 1 −11 6

 and
b =


44
−24
9

.
i) Find the general solution to the associated homogeneous equation A ∗ x = 0. Express your answer in
parametrized vector form.
ii) Find a particular solution to the original non-homogeneous equation.
iii) Use the superposition principle to express the general solution to the original equation in terms of
your answers to i) and ii).
Problem 6 Let a =

1
2

, b =

3
4

. Find a vector x such that x
T ∗a = 3 and x
T ∗b = 7. Then determine
if x is unique. [Hint: represent the problem as a system of equations]
For the remaining four problems, you are to determine of if the given statement is true or false, and provide
a short justification for your answer (for example, an argument showing it is true, or a counter-example
if it is false).
Problem 7 If A and B are non-singular 2 × 2 matrices, then A + B is also non-singular.
Problem 8 If A and B are symmetric 2 × 2 matrices, then A ∗ B is also symmetric.
Problem 9 If A is a non-singular 2 × 2 matrix satisfying the equation A2 = A, then A is the identity
matrix.
Problem 10 If A and B are 2 × 2 matrices satisfying A ∗ B = 0 (the 2 × 2 zero matrix), then either
A = 0
2×2 or B = 0
2×2
, where 0
2×2 denotes the 2 × 2 matrix with each entry equal to zero.