MTH 201: Linear Algebra

Problem Sheet 3

August 20, 2014

1. Let T be the linear operator on R3 defined by

T (x1, x2, x3) = (3x1, x1 x2, 2x1 + x2 + x3).Is T invertible? If so, find the matrix for T1.

2. Find 2 2 matrices A and B such that AB 6= BA.

3. (a) For which values of the constant k is the matrix

[2 35 k

]invertible?

(b) For which values of the constant k are all entries

[2 35 k

]1integers?

4. The cross product of two vectors in R3 is given bya1a2a3

b1b2b3

=a2b3 a3b2a3b1 a1b3a1b2 a2b1

.Consider any arbitrary vector ~v in R3. Is the transformation T (~x) = ~v~xfrom R3 to R3 linear? If so, find its matrix in terms of the components ofthe vector ~v.

5. Compute[0 0 1

] a b cd e fg h k

047

.6. Find all matrices X that satisfy the given matrix equation:1 42 5

3 6

X =1 0 00 1 0

0 0 1

.7. Show that if a square matrix A has two equal columns, then A is not

invertible.

8. Let p be a prime. Consider the set Z/pZ of integers modulo p and defineaddition and multiplication modulo p.

(a) For p = 2, 3, 5, 7 write down the addition and multiplication table forthis set.

(b) Check in these cases that Z/pZ with these operations forms a field.This field is often denoted by Fp. Observe that these fields have onlyfinitely many elements.

1