Download test 2

Linear Algebra test two
due Thursday, February 12, 2004
please show your work to get full credit for each problem
(1) Solve the system of equations
3x + 2y
4x + 5y
= 14
= 20
(a) Using the matrix inverse method.
(b) Using Cramer’s Rule.
(2) Suppose that A is a 3 × 3 matrix and that
a b c
det(A) = r s t
x y z
=6
Find the values of
(a) det(3A) (b) det(A3 ) (c) det((2A)−1 ) (d) det(AT A)
(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
b
0
0
(g) 2r + 3x 2s + 3y
x
y
(e) det(P AP −1 )
1
c
2
0
9 2t + 3z
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
3
4 7
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) Use homogeneous coordinates to write the standard matrix of the transformation T : R2 → R2 which reflects points across the line y = x + 3. (Your
answer should be a 3 × 3 matrix)
(8) The software program Matlab produced the following factorization (the socalled 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 .
(9) 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)
(10) 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
(11) ( extra credit )
~ =< a1 , a2 , a3 > , B
~ =< b1 , b2 , b3 > , & ~v =< x1 , x2 , x3 >represent vectors
Let A
3
in R .
(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 .