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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3
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))
29
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.
Consider a 3 3x matrix A given by
11 12 13
21 22 23
31 32 33
a a a
A a a a
a a a
.
Then det(A) is:
(a11a22a33 + a12a23a31 + a13a21a32) – (a11a23a32 + a12a21a33 + a13a22a31).
For a 3 3x matrix, one may think of the answer as the sum of the product of the numbers on each
of three "positive diagonals" subtracted from the sum of the product of the numbers on each of
three "negative diagonals". To visualize these "diagonals" it helps to repeat the first and second
column as follows:
11 12 13 11 12
21 22 23 21 22
31 32 33 31 32
det( )
a a a a a
A a a a a a
a a a a a
The three “positive” diagonals start with the first three entries on the first row and go
down to the right. The three “negative” diagonals start with the last three entries on the first row
and go down to the left. Generations of future engineers have used this memorization devise to
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compute determinants of 3 3x matrices. They are often quite disappointed when no comparable
visual image is available for determinants of 4 4x matrices.
For example, consider the matrix A given by
1 6 4
2 7 3 .
8 9 5
A
Then
1 6 4 1 6
det( ) 2 7 3 27
8 9 5 8 9
A
- - - + + +.
(1 7 5) (6 3 8) (4 2 9)
(4 7 8) (1 3 9) (6 2 5)
60
Alternatively, the determinant of a 3 3x matrix can be defined recursively; that is, it can
be defined in terms of determinants of 2 2x matrices. Let A be a 3 3x matrix given by:
11 12 13
21 22 23
31 32 33
a a a
A a a a
a a a
Let A ij denote the matrix obtained from A by deleting the i th row and the j th column. We
give here the definition of the determinant of A using the expansion along the first row:
11 11 12 12 13 13det (A) = a det(A ) - a det(A ) + a det(A )
22 23 21 23 21 22
11 12 13
32 33 31 33 31 32
a a a a a aa a a
a a a a a a
For example, let A be the matrix from the previous example,
A =
1 6 4
2 7 3
8 9 5
Expanding along the first row, we have
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5
11 11 12 12 13 13det (A) = a det(A ) - a det(A ) + a det(A )
6 4 1 4 1 6
1 2 7 3 6 2 7 3 4 2 7 3
8 9 5 8 9 5
4
8 5
1 6
9
7 3 2 3 2 71 6 4
9 5 8 5 8 9
= 1 [(7 5 -(9 3)] - 6 [(2 5)-(8 3)] + 4 [(2 9)-(8 7)]
= -60
We see that the determinant of a 3 3x matrix is a certain linear combination of three determinants
of 2 2x matrices. Similarly, the determinant of a 4 4x matrix is a certain linear combination of
four determinants of 3 3x matrices. More generally, the determinant of an nxn matrix is a linear
combination of n determinants of ( 1) ( 1)n x n matrices. The recursive definition of
determinants allows us to say that the determinant of an nxnmatrix is equal to a sum of many
determinants of 2 2x matrices. This method leads to the general definition of determinant of an
nxnmatrix. Let A = ( ija ) be an nxn matrix, where 2n .
Then 1+n
11 11 12 12 1n 1ndet (A) = a det(A ) - a det (A ) + ...+ (-1) a det(A )
This is called the cofactor expansion along the first row. We can obtain the same result
by expanding along any row or even any column of the matrix A . This is summarized in the
following Laplace Expansion Theorem.
The Laplace Expansion Theorem: The determinant of an nxn matrix A = ( ija ), where 2n ,
can be computed by using a cofactor expansion along the i th row:
det (A) = ai1Ci1 + ai2Ci2+...+ainCin
1
n
ij ij
j
a C ,
or a cofactor expansion along the j th column:
det (A) = a1jC1j + a2jC2j+...+anjCnj
1
n
ij ij
i
a C
where
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Cij = (-1) i+j
det (Aij)
is the ( , )i j cofactor of A .
This expansion theorem refers to the cofactor expansion along the i th row. The term
Cij = (-1) i+j
det (Aij) involves a plus or minus sign which allows for expansion along any row or
column when using summation. A quick way to determine if the sign is positive or negative is to
remember that the signs represent a checkerboard pattern:
For example, let A be the following 4 4x matrix.
13 3 4 5
0 1 2 1.
0 1 5 7
1 0 7 9
A
We want to expand along a row or column that has the most number of zeros. Since the first
column has two zeros, we can expand along the first column, and we obtain:
11 11 41 41det( ) det( ) ( )det( )
1 2 1 3 4 5
13 det 1 5 7 ( 1)det 1 2 1
0 7 9 1 5 7
13 ( 7) 30
121
A a A a A
Now that we know how to compute the determinant of nxnmatrix, let's investigate some
properties of determinants. First, we introduce a few definitions. An nxnmatrix A = (aij) is
diagonal if aij = 0 whenever i j ; in other words, a diagonal matrix is a matrix that has all zero
elements except along the diagonal. For example,
A =
11
22
33
44
0 0 0
0 0 0
0 0 0
0 0 0
a
a
a
a
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Expanding along the first row, we see that the determinant is the product of the elements on the
diagonal. Note that the identity matrix In of size n is a special case of a diagonal matrix where all
the diagonal elements are ones (and the remaining elements are all zeros). Thus det (In) = 1.
I4 = 4
1 0 0 0
0 1 0 0 det (I ) = 1
0 0 1 0
0 0 0 1
An nxn matrix A = (aij) is upper triangular if a ij = 0 for i > j (that is, all entries below the
diagonal are zero) and lower triangular if a ij = 0 for i < j (that is, all entries above the diagonal
are zero). We say a matrix is triangular if the elements in the matrix that lie above the diagonal
or lie below the diagonal are zero; that is, if the matrix is either upper or lower triangular. For
example, the following matrices A and B represent 3 3x triangular matrices; A is upper
triangular, and B is lower triangular.
3 2 1
2 1
5
0
0 0
A
1
6 2
1 4 3
0 0
0B
A diagonal matrix is considered both upper and lower triangular
Using the same ideas as in our previous example,
3 2 1
2 1
5
0
0 0
A
2 13
0 5
3 2 5
Using cofactor expansion along the first column, we have a "positive diagonal" of 2 5 and a
“negative diagonal” of 0 1 resulting in the product 3 2 5 . More generally for any triangular
matrix the determinant is the product of the elements on the diagonal. Thus we have:
Property 1: If ( )ijA a is triangular, then det (A) is the product a11a22...ann of the entries on the
main diagonal.
Property 2: The determinant changes sign when two rows are exchanged.
For example let A be a 2 2x matrix,
11 12
21 22
,a a
Aa a
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8
with
11 22 12 21det( )A a a a a
and let B be the matrix obtained by allowing permutations of the two rows of A:
21 22
11 12
a aB
a a.
Then
21 12 22 11 11 22 12 21det( ) ( ) ( ) det( ).B a a a a a a a a A
Property 3: If two rows (or columns) of a matrix are identical, then the determinant is zero.
Consider the 3 3x matrix A:
11 12 13
21 22 23
11 12 13
.
a a a
A a a a
a a a
Using the previous property (2), we know that the determinant changes sign when rows are
exchanged. But it also has to stay the same, because the matrix stays the same. The only number
that can do this is zero, so det( ) 0A .
Property 4: If a matrix contains a row (or column) of zeros, then the determinant is equal to
zero.
This is clear if we use the cofactor expansion along the zero row (or column).
The next property is about determinants of products of matrices. We first study how to
multiply two matrices. We can multiply two matrices A and B if, and only if, the number of
columns in the first matrix A equals the number of rows in the second matrix B .
For example if
13 1311 12 11 12
23 2321 22 21 22
31 32 33 31 32 33
, and
a ba a b b
a ba a b bA B
a a a b b b
are two matrices, their product is a third matrix,
Witherell
9
333231
23
13
2221
1211
ccc
c
c
cc
cc
C
where
1 1 2 2 3 3 .ij i j i j i jc a b a b a b
A concrete example is shown below. Let
2 1
5 2
4 1
A and1 2 3 4
.1 3 2 5
B
Then A is a 3 x 2 matrix and B is a 2 x 4 matrix, and we can compute the product AB, which is
a 3 x 4 matrix.
Let's explain how one gets the (2, 3) entry, 23c : the entry 23c involves the second row of
A , and the third column of B , namely 5 2 and 3
2. Then 23c = 5 3 2( 2) 11. Similarly,
we can find the new elements of the first row of the product ijC= AB= (c ) by taking the
"product" of the first row of A with each of the columns of B . The second row of C is
computed by taking the "product" of the second row of A with each of the columns of B .
Lastly, the third row of C is computed by taking the "product" of the third row of A with each of
the columns of B. We leave the details to the reader:
C = AB =
4 13
11 30 .
5 11 14 1
1
3 4
1
1
More generally, if A is an mxk matrix and B is a kxn matrix, then their product is a mxn
matrix C, where
1
k
ij it tj
t
c a b .
For example a 5 7x matrix multiplied by a 7 6x matrix yields a 5 6x matrix.
As an aside, we can also define how to multiply an nxmmatrix A by a real number t.
Any matrix, multiplied by a real number t, is a new matrix in which each entry is multiplied by t.
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10
For example, if
A=
4 3
11
2
,
then
4 3 2 4 2 ( 3)8 6
2 1 11 21 2 2 1
2 2
A
Property 5: Given nxnmatrices A and B, the determinant of the product AB is the product of
the determinants:
det( ) det( ) det( )AB A B
There is further discussion on this property in [2].
In the example given above, we can not compute the determinants of the matrices since the
matrices are not square matrices. However, if we use
A' =2 1
5 2 and B' =
1 2
1 3,
then
C' =1 1
3 4
and the determinant of C' is given by
det( ') 1 ( 1) ( 1). det( ')det( ').C A B
We say a matrix A is "invertible" if there is a matrix B such that BA = I and AB = I. We call B =1A the "inverse matrix" of A.
Property 6: The determinant of the inverse matrix1A of a matrix A is the reciprocal of the
determinant of A .
Since AA-1
= I, by property (5) we have:
det(A)det(A-1)
=det(AA-1
)=det(I) = 1.
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11
Thus
1 1det( )
det( )A
A,
Property 7: If the determinant of A is zero, then A is not invertible.
By the previous discussion, if a matrix A is invertible, then
(det 1)(det ) det 1A A I
And so, if det( ) 0,A then
10 det( ) 1A
This is impossible, because any number multiplied by zero is zero. Thus if a matrix is invertible,
then its determinant is not zero.
Using the knowledge about determinants, we will continue the investigation of
applications of determinants into the world of geometry. Consider a parallelogram with three
vertices at (0,0), (a,b), and (c,d).
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12
Consider the matrix
Aa b
c d.
Then
det(A) = ad-bc .
Let's go back to the basics and compute the area of the parallelogram shown in the illustration
above.
Let's find the area of the rectangle R with vertices at (0,0), (a+c, b+d), and (0, b+d) :
A(R) = (b+d)(a +c)
= ab +bc + ad +cd
Now we find the area of the small rectangles 1R and
2R .
1 2( ) ( )Area R bc Area R
Next we find the area of the four triangles:
1 3
1( ) ( )
2Area T ab Area T
2 4
1( ) ( )
2Area T ab Area T
Thus,
Area (Parallelogram)1 1 2( ) 2 ( ) 2 ( ) 2 ( )Area R Area R Area T Area T
1 1( ) 2 2 2
2 2ab bc ad cd bc ab cd
ad bc
Hence, we see that the determinant of the matrix A = a b
c d, whose columns contain the
coordinates of two of the parallelogram vertices, gives the area of the corresponding
parallelogram in the plane. Since we are working with the distance between points, we use the
absolute value of the difference to represent the area of the figure.
In general, the absolute value of the determinant of an appropriate 3 3x matrix can give the
volume of a parallelepiped (a slanted box).
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13
If 11 12 13u a a a , 21 22 23v a a a ,and 31 32 33w a a a equal the row elements of a
matrix, then the volume of the slanted box determined by u, v, and w (see picture) is the absolute
value of the determinant of the matrix .
u
A v
w
A three dimensional version of a two
dimensional statement can be investigated further, but a discussion of finding the volume
involves more mathematical tools than what we have available at this time. The key point is that
determinants are closely related to area and volume formulas.
Determinants can also be used to produce equations of lines and of planes. Consider two
distinct points (x1,y1) and (x2,y2). We know that we can determine the equation of the line
through those two points by finding the slope:
2 1
2 1
y ym
x x
For example, if the two points on the line are (3,1) and (5,7), then the slope of the line is:
m = (7 1) 6
3(5 3) 2
,
and the point slope equation of the line between the two points is:
(y-7)=3(x-5) 3 8y x
But we can also generate the equation of the line by setting the determinant of a 3 3x matrix A
equal to zero. Where A is given by:
A =
1
3 1 1
5 7 1
x y
Volume = det[u v w]T
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14
Let's investigate why det ( ) 0A gives the equation of the line through the two points (3,1) and
(5,7) . There is a unique line passing through these two points, and its equation has the form:
Ax + By +C = 0.
The determinant of the matrix A is:
det (A) = 2y - 6x +16
If we set det ( ) 0,A then the equation of the line is
2y = 6x + 16
y = 3x + 8
In middle school, slope intercept form is taught most often, however, we also use the
point slope formula in the classroom. Since there exists a unique line passing through the distinct
points (x1, y1) and (x2, y2), both points satisfy the equations,
slope intercept: 2 1
2 1
( ),
( )
y yy x b
x x
point slope: 2 11 1
2 1
( )( ),
( )
y yy y x x
x x
and
standard form: Ax + By +C = 0.
Most common for the algebra one classroom is the slope intercept form 2 1
2 1
( )
( )
y yy x b
x x, also
written as y mx b . Simplifying and moving everything to one side, we obtain the standard
form equation
2 1 2 1 2 1(x -x )y - (y -y )x- (x -x )b = 0 .
Notice that x is first and y is second in the standard form Ax + By +C = 0. Thus we can
simplify the equation above to obtain
1 2 1 2 1 2(y -y )x - (x -x )y+(x -x )b = 0 .
The goal is to use the point slope form 2 11 1
2 1
( )( ),
( )
y yy y x x
x xand write this in the standard
form, as we did before. We obtain
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15
2 1 2 1 1 2 1 1 2 1(y -y )x - (x -x )y-x (y -y )+y (x -x )= 0.
Expanding the term
1 2 1 1 2 1-x (y -y )+y (x -x ),
we have:
1 2 1 1 1 2 1 1-x y +x y +y x -y x =1 2 1 2-x y +y x .
Thus the equation in the standard form is
2 1 2 1 1 2 2 1(y -y )x - (x -x )y - (x y -x y ) = 0
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.
Witherell
16
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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17
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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18
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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19
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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20
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
2010 at 11:45 p.m.
[8] http://www.college-cram.com/study/algebra/matrix-algebra/determinant-of-a-2x2-matrix/
Retrieved on July 6 2010 at 11:30 p.m.
[9] http://www.mathwarehouse.com/algebra/matrix/multiply- matrix.php Retrieved on July 8
2010 at 10:30 p.m.
[10] http://www.answers.com/topic/determinant Retrieved on July 18 2010 at 11:30 p.m.
[11] http://aix1.uottawa.ca/~jkhoury/geometry.htm Retrieved on July 20 2010 at 11:30 p.m.
