MATH 262 EXAM #1 Spring 2006(SAMPLE)

1. Section 1.1

a. Find all the solutions of the following linear system

x + 2y + 3z = 1

3x + 2y + z = 1

7x + 2y − 3z = 1

b. Find all the solutions of the following linear system

x + 2y + 3z = a

x + 3y + 8z = b

x + 2y + 2z = c

where a, b, and c are arbitrary constants.

c. Find all the solutions of the following linear system7x− y = λ x

−6x + 8y = λ y

for i. λ = 5, ii. λ = 10, and iii. λ = 15.

d. Consider the linear system

x + y − z = −2

3x− 5y + 13z = 18

x− 2y + 5z = k

where k is an arbitrary constant.

i. For which value(s) of k, does this system have one or infinitely many solutions?

ii. For each value of k you found in part (i), how many solutions does this systemhave?

iii. Find all the solutions for each value of k.

2. Section 1.2

a. Solve the following system, using Gauss-Jordan elimination, for the variables x1,x2, x3, x4, and x5.

x2 + 2x4 + 3x5 = 0

4x4 + 8x5 = 0

b. Find the polynomial of degree 3 [a polynomial of the form f(t) = a+bt+ct2 +dt3]whose graph goes through the points (0, 1), (1, 0), (−1, 0), and (2,−15). Sketchthe graph of this cubic.

c. Find the polynomial f(t) of degree 3 such that f(1) = 1, f(2) = 5, f′(1) = 2, and

f′(2) = 9, where f

′(t) is the derivative of f(t). Graph this polynomial.

d. Find all vectors in R4 that are perpendicular to the three vectors1111

,

1234

,

1997

3. Section 1.3

a. Find the rank of the matrix 1 4 72 5 83 6 9

b. Compute the following dot product (if defined)

[1 9 9 7

]·

666

c. Compute the product A~x two ways: in term of the columns of A and in terms ofthe rows of A [

1 2 32 3 4

] −121

d. Compute the product A~x (if defined)[1 2 34 5 6

] [78

]

e. If the rank of a 5× 3 matrix A is 3, what is rref(A)?

4. Section 2.1

a. Find the matrix of the linear transformation

y1 = 9x1 + 3x2 − 3x3

y2 = 2x1 − 9x2 + x3

y3 = 4x1 − 9x2 − 2x3

y4 = 5x1 + x2 + 5x3

b. Consider the transformation T from R2 to R3 given by

T

[x1

x2

]= x1

123

+ x2

456

Is this transformation linear? If so, find its matrix.

c. i. For which values of the constant k is the matrix

[2 35 k

]invertible?

ii. For which values of the constant k are all entries of

[2 35 k

]−1

integers?

d. For which values of the constants a and b is the matrix

[a −bb a

]invertible?

What is the inverse in this case?

5. Section 2.2

a. Let L be the line in R3 that consists of all scalar multiples of

212

. Find the

orthogonal projection of the vector

111

onto L.

b. Let L be the line in R3 that consists of all scalar multiples of

212

. Find the

reflection of the vector

111

about the L.

c. Suppose a line L in R2 contains the nonzero vector ~v =

[v1

v2

]. Find the matrix

A of the linear transformation T (~x) = projL~x. Gives the entries of A in termsof v1 and v2.

d. Let T and L be transformations from Rn to Rn. Suppose L is the inverse of T ;that is

T (L(~x)) = ~x and L(T (~x)) = ~x

for all ~x in Rn. If T is a linear transformation, is L linear as well? Hint: ~x + ~y =T (L(~x)) + T (L(~y)), because T is linear. Now apply L on both sides.

e. Find a nonzero 2 × 2 matrix A such that A~x is parallel to the vector

[12

], for

all ~x in R2.

6. Section 2.3

a. Decide whether the following two matrices are invertible. If they are find theinverse.

A =

1 2 21 3 11 1 3

, B =

0 0 10 1 01 0 0

b. Decide whether the following the following linear transformation is invertible.Find the inverse transformation if it exists

y1 = x1 + 3x2 + 3x3

y2 = x1 + 4x2 + 8x3

y3 = 2x1 + 7x2 + 12x3

c. For which values of the constant k is the following matrix invertible?

A =

1 1 11 2 k1 4 k2

d. For which choices of the constants b and c is the following matrix invertible?

A =

0 1 b−1 0 c−b −c 0

e. For which choices of the constants a, b and c is the following matrix invertible?

A =

0 a b−a 0 c−b −c 0

f. Consider the diagonal matrix

A =

a 0 00 b 00 0 c

i. For which values of a, b and c is A invertible? If it is invertible, what is A−1?

ii. For which values of the diagonal elements is a diagonal matrix (of arbitrarysize) invertible?

7. Section 2.4

a. If possible compute the following matrix products.

i.

[1 10 1

] [1 23 4

]

ii.

[1 2 34 5 6

] [1 23 4

]

iii.[

1 2 3] 3

21

iv.

123

[1 2 3

]

v.[

0 0 1] a b c

d e fg h k

0

10

b. For two invertible n × n matrices A and B, determine which of the following

formulas are necessarily true.

i. (A−B)(A + B) = A2 −B2

ii. (ABA−1)3

= AB3A−1

iii. ABA−1 = B

iv. A−1B is invertible, and (A−1B)−1

= B−1A

v. (A + B)2 = A2 + 2AB + B2

v. A + B is invertible, and (A + B)−1 = A−1 + B−1