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MATH 51 MIDTERM 2
February 26, 2004
- Find the inverse of the matrix
A =
- Suppose A and B are 3x3 matrices, and that det(A) = 5 and det(B) = −2.
(a) Find det(AB).
(b) Find det(A−^1 ).
(c) Find det(2A).
- Let
A =
[
]
(a) Find the eigenvalues of A.
(b) Find the eigenvalues of A^10.
(c) The matrix
A =
has the number 3 as one of its eigenvalues. Find an eigenvector v that has 3 as its associated eigenvalue.
- Let T : R^2 → R^2 be the linear transformation defined by:
T
[
x y
]
[
x + y − 2 x + 4y
]
(a). Find the matrix A that represents the linear transformation T with respect to the standard basis S = {e 1 , e 2 }.
(b). Consider the basis B = {v 1 , v 2 } given by:
v 1 =
[
]
, v 2 =
[
]
Find the change of basis matrix C for the basis B. That is, find the matrix C such that v = C[v]B for all vectors v.
(c). Find the matrix B that represents the linear transformation T with respect to the basis B. 1
- Find each of the following limits, or else explain clearly why the limit does not exist.
(a). lim(x,y)→(2,3)
exy^ sin y 2 x + y
(b). lim(x,y)→(0,0)
xy x^2 + y^2
(c). lim(x,y)→(0,0)
xy sin x x^2 + 2y^2
6(a). Find a matrix A such that
A
[
]
, A
[
]
, A
[
]
(b). Let T : R^2 → R^2 be reflection across the line y = 3x. Find the matrix for T (with respect to the standard basis of R^2 .)
- A particle moves through space with velocity given by v(t) = (sin t, 2 t, 1). At time t = 0, the particle’s position is (2, 3 , 4).
(a) Find the particle’s speed at time t.
(b) Find the particle’s acceleration vector at time t.
(c) Find the particle’s position x(t) at time t.
- Calculate the following partial derivatives:
(a)
∂x
(xy^2 z + y^2 sin x + yz^5 )
(b)
∂y
sin(xyz + x^2 )
(c)
∂f ∂x
(2, 3) where f (x, y) = x^2 + xy + y^2.
(d)
∂^3 u ∂x∂y∂z
where u(x, y, z) = xy^2 z^3.
9(a). Find the determinant of the matrix
A =
c 0 3
(Your answer should be an expression involving c.)
9(b). Let U be the ball of radius 1 centered at the origin. For which values of c will U and A(U ) have the same volume?
- Suppose a particle moves with constant speed 5. Prove that at each time t, the particle’s velocity and acceleration vectors are perpendicular to each other.
2
