Amazing Applications of Determinants Expository Paper Keri Witherell In partial fulfillment of the requirements for the Master of Arts Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Jim Lewis, Advisor July 2010
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Amazing Applications of Determinants
Expository Paper
Keri Witherell
In partial fulfillment of the requirements for the Master of Arts Teaching with a
Specialization in the Teaching of Middle Level Mathematics in the Department of
Mathematics.
Jim Lewis, Advisor
July 2010
Witherell
2
This paper is about determinants. Determinants are real numbers and we can use them to
analyze a concept, such as finding area, volume, or the equation of a line. We often use numbers
in order to obtain information about something. For instance, in the medical field, doctors
measure all kinds of quantities to analyze if we are healthy or not. How much can a number tell
us?
We can arrive at a number or a solution many different ways. More than one method
works to solve a problem, for instance a problem on area. One can find the area of a
parallelogram by multiplying the base and the height together. In this paper, we explore and
investigate determinants, and some of their applications. We show how to find the area of a
parallelogram and the equation of a line through two distinct points using matrices and
determinants. Those are several examples in which information is gained by a single number, the
"determinant" of a square matrix.
Determinants preceded matrices, although matrices are taught prior to determinants in
the algebra classroom. Determinants occurred independently of matrices in the solution of many
problems. The notion of determinants has been around as long ago as 1683, when Seki, a self
taught child prodigy from the descendents of a samurai Japanese warrior family, computed what
we know now as determinants. Seki was able to find determinants of 2 2, 3 3, 4 4 and
5 5 matrices and applied them to solving equations.
One of the first uses of the word determinant appeared in 1748 in Maclaurin's document
Treatise on Algebra. The actual word "determinant" was not created until 1801, when it was
used by Gauss. Gauss introduced this word, but not for what we use it today, but for the
discriminant of a quartic, (a quartic is a polynomial of degree four; for example, x4). Recall that
a discriminant is an expression which gives information about the nature of the roots of
polynomials. In 1812, Cauchy made the first use of determinants in the current sense; in fact,
Cauchy was responsible for developing much of the early theory of determinants.
The determinant of a square matrix is a real number associated to the matrix. As we shall
see, the value of this number tells us whether a matrix is invertible or not, and can be used to
determine the solution of a system of n linear equations in n unknown (Cramer's rule).
Consider a 2 2x matrix A given by
A = 11 12
21 22
a a
a a,
where aij represents the entry in the i-th row and j-th column. The determinant of A is a number
defined by:
11 12
21
11 2 21
2
2
2
12det( )a a
Aa a
a aa a .
Note that 11 12
21 22
a a
a a is just a notation for the determinant of A .
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One way to visualize this process is to think of the determinant as the number obtained by
subtracting the product of the elements in the “negative diagonal” (that is, 21 12a a ) from the
product of the elements in the “positive diagonal” (that is, 11 22a a ).
As we have seen before, we usually denote the entries of a matrix with letters and subscripts. It is
interesting to note that Leibniz used subscripts to note coefficients and unknowns in a system of
equations; for example, in ija , i denotes the equation in which it occurs, and j denotes the
unknown of which ija is the coefficient.
Consider the 2 x 2 matrix A :
2 3
5 7A .
Then
2 3det
5 7A
(2 7) (5 ( 3))
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The determinant of a 3 3x matrix can be computed in two different ways. The first
method is similar to the method used for computing the determinant of a 2 2x matrix.
Note in the language of determinants the left hand side is the determinant of matrix A, where
1 1
2 2
1
1
1
x y
A x y
x y
In fact, by expanding along the first row of A, we see that
1 1 1 2
2 2 1 2
1 1det( )
1 1
y x x xA x y
y x y y
1 2 1 2 1 2 2 1( ) ( ) ( )y y x x x y x y x y
Thus we see that the equation of the line can be written as
det( ) 0A .
Mathematics processes that look different such as finding an equation of a line using
slope intercept and finding an equation of a line using language of determinants both produce the
same line. The connection is the matrix that can be built whose determinant represents the
standard form of an equation. As middle level math teachers, we can represent concepts we are
already teaching and incorporating into our classroom through the use of determinant
vocabulary.
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For example, consider the two points (3, 1) and (5,7). We write A(x,y) to emphasize that
the matrix is a function of x and y. Then the matrix described above is
1
( , ) 3 1 1 .
5 7 1
x y
A x y
Then the equation of the line through (3,1) and (5,7) is:
det (A(x, y)) = 0 2y - 6x +16 = 0.
We say three points (x1, y1), (x2, y2), (x3, y3), are collinear if they lie on the same line. If
det(A(x,y))=0 is the equation of the line through the distinct points (x1, y1), (x2, y2) as described
above, then (x3, y3) lies on the same line if and only if it satisfies the equation of the line, that is
det(A(x3, y3))=0.
Consider the matrix
We can substitute the values of (x3, y3) for x and y . If the values of (x3, y3) continue to give a
zero determinant, then the point (x3, y3) is collinear with the other two points.
For example, are the three points (5,7), (3,1) and (-1,-1) collinear? The points (5,7), (3,1) and
(-1,-1) are collinear if and only if
Since det( ( 1, 1)) 20 0,A (-1,-1) does not lie on the line through the points (3,1) and (5,7).
We can also give the following intuitive explanation: If the points in the plane are collinear, then
the parallelogram formed using any two of them, as on page 11, has zero area. Note that
expanding along the third column of the matrix A gives that the determinant is det(A13)-
det(A23)+det(A33). This is a sum of the areas of these parallelograms, so if the points are
collinear, the determinant must be zero."
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If we graph the points ( 1, 1), (3,1),C A and (5,7),D then we also see geometrically that
the points are not collinear. The graph is shown below.
If we graph the points, (4,4)B then we see geometrically that the points are collinear. Since
det( (4,4)) 0,A then (4,4) lies on the same line as points (3,1) and (5,7).
Just as we worked out for points in a plane, comparable facts are true for the planes in
space. Given three points, setting the determinant of a particular matrix to zero generates the
equation of a plane. In the capstone course, we were introduced to the idea that the graph of
linear equations in three unknowns is a plane. The technique we used to make lines in a plane
can be used to represent a plane in space.
Only one plane can pass through three non collinear points. Note that just like we can
have an infinite number of lines that meet at one point, we can have an infinite number of planes
that meet at one line. The equation of a plane is:
ax + by + cz = d
If we know the coordinates of three points (x1,y1, z1), (x2,y2, z2), (x3, y3, z3) that lie on the plane,
then we have three equations in the three unknowns a, b, c, namely
1 1 1 1
2 2 2 2
3 3 3 3
a x b y c z d
a x b y c z d
a x b y c z d
Let ( , , )A x y z be the coefficient matrix of the system above,
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We are claiming that the previous methods of finding an equation of a line can be extended to finding the equation of a planes. So the equation of a plane through three distinct points is
det( ( , , )) 0A x y z
This is an outlandish amount of work, 24 products. The amount of mathematics and computation
needed to find the equation of a plane using determinants is beyond the scope of this paper.
However, it can be further investigated with more time and mathematics. We can apply this
method to the following example.
Consider the plane going through the points (1, 1,3),(0,1,7), and (4,0, 1).
Then matrix A is
1
1 1 3 1( , , ) ,
0 1 7 1
4 0 1 1
x y z
A x y z
and the equation of the plane is given by:
1
1 1 3 1det ( , , ) 0
0 1 7 1
4 0 1 1
x y z
A x y z
.
Simplified, the equation of the plane through the three points is
As a middle level educator, I see that finding determinants is beneficial and a practical
application. The applications of computing determinants of 2 2x and 3 3x matrices, finding the
area of a parallelogram and an equation of a line are obtainable at the middle level. One example
of why determinants do not make practical sense in middle level education is that the larger nxn
matrices are more difficult and time consuming the application becomes. For instance if we have
an nxn matrix, we have n! products to compute in order to find the determinant. Even the fastest
computers cannot calculate the determinant of a large matrix using cofactor expansion. Consider
a 20 X 20 matrix; to compute its determinant would require 20! operations that is about 182.4 10x
.0417812 zyx
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operations. If a computer were able to perform a trillion operations per second, it would take 2.4
X 106 seconds, over six months, to finish the calculations.
The Math in the Middle program has shown again and again that there is more than one
way or method to solve a problem. Concepts that appear to be separate of one another may in
fact have connections. Examples have demonstrated that information can be gained by a single
number, the determinant of a square matrix. For example, as we showed in the case of an
equation of a line, using the determinant of a 3 3x matrix gives an alternative way of
representing knowledge we already had. We also showed how to find the area of a parallelogram
using prior knowledge (multiplying the base and height together), and then using new knowledge
to find the area with (determinants). In spite of everything an amazing amount of information is
packed into a determinant.
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Bibliography
[1] Leon, S.J.(1994). Linear algebra with applications, New Jersey: Prentice Hall.
[2] Poole,D. (2006). Linear algebra a modern introduction, second edition, Belmont: Thompson
Higher Education.
[3] Strang, G. (1988). Linear algebra and its applications, third edition, New York: Harcourt
Brace Jovanovich Inc.
[4] Williams, G. (1984). Linear algebra with applications, Boston: Allan and Bacon, Inc.
Web Sources
[5] http://www.intmath.com/Matrices-determinants/1_Determinants.php Retrieved on June 23
2010 at 9:30 p.m.
[6] http://www.maths.surrey.ac.uk/explore/emmaspages/option1.html#Det Retrieved on June 23
2010 at 10:00 p.m.
[7] http://www.intmath.com/Matrices-determinants/1_Determinants.php Retrieved on June 23