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Determinants and Solutions of Linear Systems of
Equations
Megan Zwolinski
February 4, 2004
Contents
1 Introduction 1
2 Determinants 1
3 An nxn Matrix 1
4 Properties of Determinants 2
5 Rules for Determinants 2
6 2 X 2 Matrix 2
7 Example 1 3
8 3 x 3 Matrix 3
9 Example 2 3
10 Solution of Linear Systems of Equations 3
11 Cramer’s Rule 4
12 The Alternative Theorem 4
1 Introduction
In this paper, we will study determinants and solutions of linear systems of
equations in some detail. We will learn the basics for each and expand on them.
2 Determinants
A determinant is a mathematical object which is very useful in the analysis and
solution of systems of linear equations. Determinants are only defined for square
matrices. A square matrix has horizontal and vertical dimensions that are the
same (i.e., an nxn matrix). The difference between the form of a matrix and a
determinant of a matrix is that a determinant is displayed using straight lines
in-place of the square brackets. The determinant is a scalar quantity, which
means a one-component quantity. The determinant is most often used to:
- test whether or not a matrix has an inverse
- test for linear dependence of vectors (in certain situations)
- test for existence/uniqueness of solutions of linear systems of equations
3 An nxn Matrix
In an nxn matrix, we follow certain rules for the appearance of the matrix.
We let vi symbolize the i
th row. (ai 1 , ai 2 , ..., ain). If this row were to be mul-
tiplied by α, the the row would appear to be (αai 1 , αai 2 , ..., αain). Also, if
two rows were added, the i
th row and the j
th column, we would have (ai 1 +
aj 1 , ai 2 + aj 2 , ..., ain + ajn). A unit matrix appears with the following rows:
(1,0,0,...,0),(0,1,0,...,0),...,(0,0,0,...,1). Also, the letters e 1 , ..., en describe a unit
row.
In an nxn matrix there are a few forms in which a determinant is recognized.
First of all the determinant symbolizes the function of the n
2 variables aij (i, j =
1 , 2 , ..., n). The determinant for this function can be written as:
D =
a 11 a 12... a 1 n
a 21 a 22... a 2 n
. . .
an 1 an 2... ann
= |aij | = D(v 1 , v 2 , ...vn)
4 Properties of Determinants
- D(v 1 , v 2 , ..., vi, ..., vn) = D(v 1 , v 2 , ..., vi + vj , ...vn) (i 6 = j) (Invariance)
- D(v 1 , v 2 , ..., αvi, ..., vn) = αD(v 1 , v 2 , ..., vi + vj , ...vn) (Homogeneity)
- D(e 1 , e 2 , ..., en) = 1 (Normalization)
5 Rules for Determinants
Determinants are either written as |A| or detA. Now say that we delete the
i
th row and the j
th column a (n − 1)x(n − 1) submatrix Aij is formed. The
8 3 x 3 Matrix
The next kind of matrix studied is the 3x3 matrix. The form of this matrix is
B =
a b c
d e f
g h i
The determinant of a 3x3 matrix is found using the following formula:
|B| = det(B) =
a b c
d e f
g h i
= a
e f
h i
− b
d f
g i
d e
g h
9 Example 2
3 x 3 Matrix Using the matrix
B =
the determinant would be
|B| = det(B) =
10 Solution of Linear Systems of Equations
Consider the system of n linear equations in n unknowns x 1 , x 2 , ..., xn
n ∑
j=
aij xj = bi (i = 1, 2 , ..., n)
11 Cramer’s Rule
The above formula is Cramer’s Rule. It states that if |A| = |aij | 6 = 0, then
n ∑
j=
aij xj = bi (i = 1, 2 , ..., n)
possesses a unique solution given by
xr =
∑n
i=
A
∗ ir bi
|A|
(r = 1, 2 , ..., n)
Cramer’s rule is used to solve a set of n linear equations in n unknowns. It uses
determinants to obtain the solution.
12 The Alternative Theorem
This formula that results from Cramer’s Rule is the Alternative Theorem. Also
described as the homogeneous system
n ∑
j=
aij xj = 0 (i = 1, 2 , ..., n)
possesses a non-trivial solution (i.e., a solution other than x 1 = x 2 = ... =
xn = 0) if and only if |A|=0. If for a fixed A = (aij ) there are solutions to the
non-homogeneous system
n ∑
j=
aij xj = bi (i = 1, 2 , ..., n)
for every selection of the quantities bi, then |A| 6 = 0 and the homogeneous system
had only the trivial solution.
