PILOT PROBLEM SET 4
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1. Inverse matrices
Exercise 1.1. Suppose that
−1
A=
2
0
,
−3
−2
B=
.
−5
(a) Show that A is invertible and find the inverse matrix of A.
(b) Find X such that AX = B.
Exercise 1.2. (a) Show that if X = AX + D, then
X = (I − A)−1 D
provided that I − A is invertible.
(b) Suppose that
3 2
−2
A=
, D=
.
0 −1
2
Compute (I − A)−1 and use your result in (a) to compute X.
Exercise 1.3. Suppose that
2 4
.
3 6
Compute det(A). Is A invertible?
Suppose that
x
b
X=
, B= 1 .
y
b2
Write AX = B as a system of linear equations.
Show that if
3
B=
9/2
then AX = B has infinitely many solutions. Graph the two straight lines
associated with the corresponding system of linear equations, and explain
why the system has infinitely many solutions.
Find a column vector
b
B= 1
b2
so that AX = B has no solutions.
A=
(a)
(b)
(c)
(d)
Exercise 1.4. Suppose that
a 8
A=
,
2 4
X=
x
,
y
B=
b1
.
b2
(a) Show that when a 6= 4, AX = B has exactly one solution.
1
2
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(b) suppose a = 4. Find conditions on b1 and b2 such that AX = B has (i)
infinitely many solutions and (ii) no solutions.
(c) Explain your results in (a) and (b) graphically.
2. Graphical representation
Exercise 2.1. Suppose a vector
x1
x2
◦
has length 4 and is 70 counterclockwise from the negative x2 -axis. Find x1 and
x2 .
Exercise 2.2. Find x + y for the given vectors x and y. Represent x, y, and x + y
in the plane,
how you add x and y.
and
explain graphically
−1
3
(a) x =
and y =
.
0 2 −3
−2
.
(b) x =
and y =
−1
3
Exercise 2.3. Let
3
u=
,
4
−1
v=
,
−2
1
w=
.
−2
(1) Compute u + v + w and illustrate the result graphically.
(2) Compute v − 21 u and illustrate the result graphically.
(3) Let v0 denote the transpose of v. Compute the 2 × 2 matrix A = uv0 . Is A
invertible?
(4) Let w0 denote the transpose of w. Compute the 2 × 2 matrix B = vw0 . Is
A invertible?
−1
Exercise 2.4. (a) Use a rotation matrix to rotate the vector
counterclockwise
2
by the angle π/6.
−2
(b) Use a rotation matrix to rotate the vector
counterclockwise by the
−3
angle π/9.
1
(c) Use a rotation matrix to rotate the vector
counterclockwise by the
−2
angle π/3.
Department of Mathematics, Johns Hopkins University, 3400 N Charles Street, Baltimore, MD 21218, USA
E-mail address: [email protected]