Factoring 2x2 Matrices with Determinant of ±1 and Integer Elements
Ryan Altfillisch
Carthage College
[email protected]
April 14, 2014
Abstract
We prove that all 2x2 matrices, as long as they meet our specifications,
have a dominant column in absolute value and can be factored to a
product of these three matrices,
and
.
1 Introduction
In this study, we will show a pattern of factoring specific matrices. These matrices will be 2x2
matrices with integer elements and a determinant of positive or negative one. We will prove
multiple theorems that will help show that all matrices fitting our specifications can be factored
into a product of three unique matrices.
For the basis of this study, let be a 2x2 matrix with integer elements and a determinant of ±1.
There can be anywhere from zero to four elements that are negative integers in . We will show
that one of
can be written as a product of the matrices
up to the facts that
and
,
and
, the identity matrix.
2 Definitions and Development
We will begin with important definitions and examples.
Definition 1 A matrix is a rectangular array of numbers. The numbers in the array are called
the elements of the matrix.
Example 2 The array
is a matrix. The numbers 0, 1, 1, and 0 are the elements of the
matrix.
Definition 3 Let be a square matrix. The determinant, denoted by
difference of the products of the two diagonals of the matrix.
Lemma 4 (Determinant of any 2x2 Matrix) If
is a 2x2 matrix
or
, then
, is the
is
.
Definition 5 If is an
matrix and is an
matrix, then the product
is the
matrix whose elements are found as such. To find the element in row and column of
,
multiply the corresponding elements of row of and column of and adding the products.
Example 6 The product of
and
is
.
Definition 7 The identity matrix, denoted as , is a square matrix with 1’s in main diagonal and
0’s elsewhere. The product of and any matrix is .
Example 8 The 2x2 identity matrix is
.
Definition 9 Let be a square matrix. Then, if a matrix
then is called the inverse of .
Example 10 Given that
can be found such that AB  BA  I ,
is a square matrix, then the matrix
inverse of , as seen by
.
Definition 11 A matrix
. Also,
and
is the

has a dominant column if
has a dominant column in absolute value if
and
or
and
and
or
.
Example 12 The matrix
Note that
has a dominant column because
, and
Example 13 The matrix
Note that
and
.
.
has a dominant column in absolute value.
, and
Example 14 The matrix
Note that
.
does not have a dominant column in absolute value.
, and
.
The next two theorems were proved by Kim Mineau in her senior thesis at Carthage College.
Theorem 15 Every 2x2 matrix
with non-negative integer entries and a determinant of
±1 must have a dominant column, except and
and
. That is either
and
or
[1].
Theorem 16 Every 2x2 matrix with non-negative integer entries and a determinant of ±1, except
for , can be factored uniquely into a product of
and
[1].
3 Results
Next, we will expand upon Theorems 15 and 16 to include all matrices with integer elements.
Theorem 17 (Dominant Column in Absolute Value Theorem) Every 2x2 matrix
with integer elements and a determinant of ±1 must have a dominant column in absolute value,
except the identity matrix,
and
and
. That is, either
and
or
.
Proof. There are 5 different cases to prove.
Case 1: The matrix A has one negative element on the main diagonal. Remember that
. If
with
and
, which implies that –
and
either
, then we know that
, which also implies that
or
and
with
, or
. If
, then
with
dominant column in absolute value, where either
If
. This means that either
with
. If
, then
. In each case, there is always a
and
or
and
.
a similar argument can be made that always ends with a dominant column in
absolute value.
Case 2: The matrix A has one negative element not on the main diagonal. If
and
. We know that
means that either
. If
and
, then
with
, which implies that
or
and
where
or
. If
. This
, then
where
where
. In each case, the
matrix always has a dominant column in absolute form. A similar argument can be made if
that always ends with a dominant column in absolute form.
Case 3: The matrix A has two negative elements. If
or
, then
and
, respectively, are now positive matrices, with the elements in the same positions.
Therefore, according to Theorem 15, there must be a dominant column in absolute value. Let us
assume that
where
and
, and suppose that
and
.
This implies that
. Also, we know that
and
. We know that
, which implies that
. Since
and
generates , and
and
generates
, we will assume that either
or
and either
or
. Then,
. This implies that
, which implies that
, which is a contradiction.
Thus, not having a dominant column in absolute value causes to not have a determinant of
Case 4: If matrix A has three negative elements, then multiplying A by either
under case 1.
will make it fall
Case 5: If matrix A has four negative elements, then A is a positive matrix and, from Theorem
15, must have a dominant column in absolute value.

Therefore, in each case, we see that there always ends up being a dominant column in absolute
value. 
Theorem 18 (Factorization Theorem) Every 2x2 matrix with negative elements and a
determinant of ±1, except for the identity matrix, can be factored into a product of
, and
a product of
. Each product will be in the form
and
,
where
and
is
.
Proof. We know that we must have a dominant column in absolute value by Theorem 17. Note
that there are 5 cases to prove.
Case 1: The matrix
has one negative element. If
1, we know that
multiply
by
or
or
where
in the first case we will get
get
, then from Theorem 17, Case
in each case. If we
by –
. Then if we multiply
we will
which is now a positive matrix. In the second case, if we multiply
by
we will get
we will get
which is a positive matrix. In the third case, if we multiply
which is also a positive matrix. If
, and then multiply
by . Then we will get
, then if we multiply
reasoning. If
, then we will multiply
by
which will follow the same logic. If
by , we will get
then we will multiply
by ,
which follows the same
by , and we will get
. This will
now follow the same thought.
Case 2: The matrix A has two negative elements. If
is now a positive matrix. If
, then
If
by
, then we will multiply
, we will get
multiply
by
, then
, and there’s now a positive matrix.
, then
. Then if we multiply
which is now a positive matrix. If
and will get
, and there
. Then, if we multiply
by
, then we will
by
, we will get
, which is now a positive matrix.
Case 3: The matrix has three negative elements. If we multiply
by , then we would have
a matrix with one negative element which will follow the same logic as case 1.
Case 4: The matrix
has four negative elements, namely
, then
,
and there’s now a positive matrix.
Thus, in each case the original matrix
factored into the form
. 
is now positive and will follow Theorem 16, and will be
Example 19 We will factor the matrix
.
The matrix F has two negative elements, therefore we can multiply by
. For this example we
will choose w. The product matrix has a dominant left column and we will multiply by
. The
product matrix has a dominant left column and we will multiply by
. The product matrix has a
dominant left column and we will multiply by
. The product matrix has a dominant right
column and we will multiply by .
Therefore,
.
Conjecture 20 (Uniqueness) If we were given any random combination of , , and , such as
, can we write that in the form of
? We believe that the answer is yes.
For this example, the combination of
so
can be written as
. There are a number of different combinations that can cancel any combination of
, , and , such as
,
and
. Using these combinations, along
with others that exist, will take any combination of , , and where is not on the end and
write it uniquely on the end.
4 Conclusion and Directions for Further Research
We have proven that there are three distinct matrices that generate all 2x2 matrices with
determinant ±1 with negative entries. This is interesting to the point that a few simple matrices
can generate such a large group of matrices. In the future, we can see if there is a way to get rid
of negatives occurring in the factorization, whether this means adding in a fourth generator or
changing one of the generators. Also, we can prove that any combination of , , and can be
written uniquely in the form
.
References
[1] Mineau, K., Factoring 2x2 Matrices with Determinant of ±1, Carthage College, Kenosha, WI, 2012.
[2] Williams, G., Linear Algebra with Applications, Wm. C. Brown Publishers, Dubuque, IA, 1991.