Download test 2

Linear Algebra test two
due Monday, November 8, 2004
please show your work to get full credit for each problem
(
1. Solve the system of equations
3x + 2y = 14
4x + 5y = 20
)
(a) Using the matrix inverse method. (~x = A−1~b )
(b) Using Cramer’s Rule.
2. Suppose that A is a 3 × 3 matrix and that
a
det(A) = r
x
b c s t = 5
y z Use this information and the properties of the determinant to find the values of
(a) det(3A) (b) det(A3 ) (c) det((2A)−1 ) (d) det(AT A) (e) det(P AP −1 )
(in part (e) assume that P is an invertible 3 × 3 matrix)
a − 2x
x
(f) r
b − 2y
y
s
c − 2z z
t
a
0
(g) 2r
+
3x
x
b
1
c
0
2
0
2s + 3y 9 2t + 3z
y
14
z
(h) Find the value of the matrix entry (A−1 )2,3
(i) If S is the unit ball {(x1 , x2 , x3 ) ∈ R3 : x2 + y 2 + z 2 ≤ 1}, find the volume of the image
of the unit ball under multiplication by the matrix A.


2
4
2

3. Find the LU factorization of the matrix A = 
 −6 −9 −5 
8 10 10
"
4. Given the matrix A =
2 4 7
3 5 8
#
(a) Is it possible to find a matrix B so that with matrix multiplication,
AB = I2×2 ?
Here I2×2 is the 2 × 2 identity matrix. If it is possible, produce such a matrix B.
(b) Is it possible to find a matrix C so that with matrix multiplication,
CA = I3×3 ?
Here I3×3 is the 3 × 3 identity matrix. If it is possible, produce such a matrix C.
page two
5. Is it possible to find a 3 × 3 matrix E with 2 pivots and with det(E) = 12 ? If so, produce
such a matrix E.
6. Is it possible to produce a 2 × 2 matrix F with diagonal terms F1,1 = 2 & F2,2 = 1 such
that det(F −1 ) = 12 ? If it is possible, produce such a matrix F .
7. The software program Matlab produced the following factorization (the so-called singular
value decomposition) for the 2 × 2 matrix A on the left side of the equation into the matrix
product of U , S, & V on the right side of the equation:
A = U SV
"
1 2
0 1
#
"
=
0.9239 −0.3827
0.3827
0.9239
#"
2.4142
0
0 0.4142
#"
0.3827 −0.9239
0.9239
0.3827
#
(a) Describe the geometric effect of each of the matrices U , S, & V . (Hint: U & V are
rotation matrices–find the angle of rotation they correspond to)
(b) Describe the relationship(s) between the matrices U & V .
8. Find the volume of the tetrahedron bounded by the points A(1, 3, 5), B(−2, 5, 7),
C(5, 9, 17), & D(−5, −1, 2). (Hint: Use the fact that the volume of a tetrahedron is one-sixth the
volume of the parallelepiped with which it shares three edges)
9. Explain why each of the following statements is either true or false.
(a) If A is a 3 × 3 matrix and {~v1 , ~v2 , ~v3 } is a linearly dependent set of vectors in R3 , then
{A~v1 , A~v2 , A~v3 } is also a linearly dependent set.
(b) If A is a 3 × 3 invertible matrix and {~v1 , ~v2 , ~v3 } is a linearly independent set of vectors
in R3 , then {A~v1 , A~v2 , A~v3 } is also a linearly independent set.
(c) If A is a 3 × 3 matrix and {~v1 , ~v2 , ~v3 } is a linearly independent set of vectors in R3 for
which {A~v1 , A~v2 , A~v3 } is also a linearly independent set, then the matrix A must be
invertible.
(d) If A is a 3×3 matrix and {~v1 , ~v2 } is a linearly independent set of vectors in R3 for which
{A~v1 , A~v2 } is also a linearly independent set, then the matrix A must be invertible.
(e) If A and B are 3 × 3 matrices and the product AB is known to be invertible, then it
follows that B is also invertible.
page three
10. ( extra credit )
~ =< a1 , a2 , a3 > , B
~ =< b1 , b2 , b3 > , & ~v =< x1 , x2 , x3 >represent vectors in R3 .
Let A
(a) Find the standard matrices for the linear transformations
~ × ~v
L(~v ) = A
~ × ~v
T (~v ) = B
where the operation on the right-hand side of each equation is the vector cross-product.
Call these standard matrices MA and MB , respectively.
(b) Show that the standard matrix for the linear transformation
~ × B)
~ × ~v
P (~v ) = (A
is equal to MA MB − MB MA .