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Unit 1
Basic Elements on Linear Algebra
and Matrix Theory
1.1. Representation of Economic Data through Re-
al Matrices. Types of Matrices and Matrix
Operations
Definition 1.1.1 A real matrix A of order or size m × n is a rectangular array
of m · n real numbers arranged in m rows and n columns. Every a ij is called the
(i,j)-entry of the matrix A.
Notation 1.1.2 The set of all real matrices of size m × n is denoted either by
M
m×n (IR) or by M m,n (IR). Thus A ∈ M m×n (IR) denotes that A is a real matrix of
size m × n, and A can be represented by:
A =
a 11 a 12 · · · a 1 n
a 21 a 22 · · · a 2 n
a m 1 a m 2 · · · a mn
= (a ij
i = 1,... , m;
j = 1,... , n.
1.1.1. Types of Matrices
Definition 1.1.3 The zero matrix 0 is the matrix whose entries are all 0 (that
is, a ij = 0, ∀i = 1,... , m, ∀j = 1,... , n).
Definition 1.1.4 A row matrix is a matrix containing only one row or, conse-
quently, a 1 × n matrix.
Definition 1.1.5 A column matrix is a matrix containing only one column or,
consequently, an n × 1 matrix.
4 UNIT 1. BASIC ELEMENTS ON LINEAR ALGEBRA AND MATRIX THEORY
Definition 1.1.6 A square matrix is an n × n matrix.
Remark 1.1.7 If A is a square matrix of size n × n, it can also be said that the
size of A is n.
Notation 1.1.8 The set of all square matrices of size n is denoted by M n
(IR).
Definition 1.1.9 Let A = (a ij
) ∈ M
n (IR) be a square matrix of size n. Then:
- A is said to be a diagonal matrix if a ij = 0, ∀i 6 = j. The a ii entries form the
main diagonal of A.
- A is the identity matrix of size n if A is a diagonal matrix and a ii
∀i = 1,... , n (that is, every entry is zero except those entries in the main
diagonal whose value is 1). The identity matrix of size n is denoted by In or
Id n
- A is said to be symmetric if a ij = a ji , ∀i, j = 1,... , n.
- A is said to be skew-symmetric if a ij = −a ji , ∀i, j = 1,... , n.
- A is said to be upper triangular if a ij = 0, ∀i > j (that is, all entries below
the main diagonal are zero).
- A is said to be lower triangular if a ij = 0, ∀j > i (that is, all entries above
the main diagonal are zero).
Definition 1.1.10 Let A = (a ij
) ∈ M
m×n (IR) be an m × n matrix. A submatrix
of A is defined as any k × r matrix obtained by removing m − k rows and n − r
columns from A.
1.1.2. Matrix Operations
Definition 1.1.11 (Matrix Addition) Let A, B ∈ M m×n (IR) be two real matri-
ces of the same size. Then, the sum matrix of A and B, denoted by A + B, is a
new matrix C ∈ M m×n (IR) obtained as follows:
c ij = a ij
- b ij , ∀i = 1,... , m; ∀j = 1,... , n.
Definition 1.1.12 (Scalar Multiplication) Let λ ∈ IR be a real number and let
A ∈ Mm×n(IR) be a matrix. The product of the scalar λ by the matrix A,
denoted by λ · A, is a new matrix C ∈ M m×n (IR) obtained as follows:
c ij = λ · a ij , ∀i = 1,... , m; ∀j = 1,... , n.
6 UNIT 1. BASIC ELEMENTS ON LINEAR ALGEBRA AND MATRIX THEORY
1.1.4. Rank of a matrix
The rank can be defined and computed for matrices of any size, either square
or not. Previously, the concept of minor of size r (with r ≤ min{m, n}) of a matrix
A ∈ M
m×n (IR) must be defined.
Definition 1.1.19 Let A ∈ M m×n (IR) be an m × n matrix and let r be a natural
number so that r ≤ min{m, n}. A minor of size or order r is the determinant of
a square submatrix of size r of A.
Definition 1.1.20 Let A ∈ M m×n (IR) be an m × n matrix. The rank of A is the
largest size of a nonzero minor of A. It is denoted by rank(A) or rk(A).
Remark 1.1.21 Note that the minor whose size equals the rank of the matrix can
be unique or not, but every minor of a larger size must vanish.
There are several methods to compute the rank of a matrix, but we will only
describe here the frame method:
Step 1. If A is a zero matrix, rk(A) = 0. Otherwise, find a nonzero element of
A. This element is itself a nonzero minor of A (of size 1). So, rk(A) is at least
r = 1.
Step 2. Add to the nonzero minor (of size r) previously obtained a row and
a column both different from those included in the minor (of size r). So we
obtain a minor of size r + 1.
Step 3. If the previously chosen minor of size r + 1 is nonzero, then add 1 to
r (r → r + 1), note that rk(A) is at least this new r (the old r + 1), and run
again Step 2. Otherwise, go to Step 4.
Step 4. If all the rows and columns have already been added to the minor
(of size r) and none of these minors (of size r + 1) is nonzero, rk(A) = r and
the method ends. Otherwise, change the row or the column and obtain a new
minor (of size r + 1) from the nonzero minor (of size r) and the new row and
column, and run again Step 3.
1.1.5. Inverse of a Square Matrix
Definition 1.1.22 A square matrix A ∈ M n (IR) of size n is called invertible or
nonsingular if a matrix B ∈ Mn(IR) exists, so that the two following conditions
are satisfied: A · B = I n and B · A = I n
The matrix B verifying the above property is called the inverse matrix of A.
Notation 1.1.23 If there exists, the inverse matrix of A is denoted by A
− 1 .
VECTORS. LINEAR DEPENDENCE AND INDEPENDENCE 7
Proposition 1.1.2 The inverse of a square matrix, if it exists, is unique.
Proposition 1.1.3 Let A ∈ M n (IR) be a square matrix of size n. A is invertible if
and only if |A| 6 = 0.
The problem of finding inverse matrices can be reduced to apply a certain math-
ematical formula. However, to state this, a previous concept is needed: the concept
of cofactor matrix.
Definition 1.1.24 Let A = (a ij
) ∈ M
n (IR) be a square matrix of size n. Then:
- The minor of the a ij entry or the (i,j)-minor, denoted by α ij , is the de-
terminant of the square submatrix obtained by removing the i
th row and the
j
th column from A.
- The cofactor of the a ij entry or the (i,j)-cofactor is the real number
A
ij
i+j · α ij
- The cofactor matrix of A is the matrix C(A) = (A ij
Proposition 1.1.4 If A ∈ M n (IR) is an invertible matrix, then the inverse matrix
of A, the A
− 1 matrix, can be computed by using the following formula:
A
− 1
|A|
· C(A)
t .
1.2. Considering Variables of Several Dimensions.
Vector Operations. Linear Dependence and
Linear Independence
Definition 1.2.1 Let n ∈ IN be a nonnegative integer. A vector with n compo-
nents is any column matrix v of size n × 1. Every entry of the matrix v is called
component or coordinate of the vector v.
Notation 1.2.2 The set of all n-component vectors is denoted by IR
n
. Thus, v ∈ IR
n
means that v is an n-component vector.
Remark 1.2.3 This is not the most general definition for the concept of vector, but
it will be used along this subject, because it is quite easy to understand. However,
sometimes through the course, vectors of IR
n will be treated as row matrices instead
of column matrices.
1.3. LINEAR MODELS OF SEVERAL EQUATIONS. SOLVING AND INTERPRETING SYSTEM
1.3. Linear Models of Several Equations. Solv-
ing and Interpreting Systems of Linear Equa-
tions
Let us consider a system of m linear equations with n unknowns:
a 11 x 1
a 21 x 1 + · · · + a 2 nxn = b 2
am 1 x 1 + · · · + amnxn = bm
where a ij , b i and x j are real numbers, for all i ∈ { 1 ,... , m} and for all j ∈ { 1 ,... , n}.
Definition 1.3.1 In the above system, the scalars aij are called the coefficients
of the system; b i are the constant terms (commonly called right-hand side)
and xj are the unknowns.
Linear systems can also be written using matrices. This way of writing is called
the matrix form of the system. The matrix form of the above system is the
following:
A · x = b,
where:
A =
a 11 · · · a 1 n
am 1 · · · amn
,^ x^ =
x 1
xn
,^ b^ =
b 1
bm
So, A is the coefficient matrix of the system; x is the unknown vector and b is
the right-hand side vector.
1.3.1. Discussion of a System of Linear Equations
Definition 1.3.2 A solution of the system A · x = b is any vector x 0
∈ IR
n that
simultaneously verifies the m equations of the system (that is, A · x 0
= b).
According to the number of solutions, three distinct types of linear systems can
be found:
- Inconsistent System (I.S.): a system with no solutions.
- Consistent System: a system having at least one solution. There are two
subtypes of consistent systems:
a) Uniquely Determined System (U.D.S.): it has exactly one solution.
10 UNIT 1. BASIC ELEMENTS ON LINEAR ALGEBRA AND MATRIX THEORY
b) Underdetermined System (U.S.): it has more than one solution. In
this case, the system has infinitely many solutions.
Definition 1.3.3 The augmented matrix of the linear system A · x = b is the
matrix obtained when adding the column b to the right side of A. It can be denoted
by A
∗ as well as by (A|b).
Remark 1.3.4 Notice that if the coefficient matrix of the system A is of size m×n,
then the augmented matrix is of size m × (n + 1).
Theorem 1.3.5 (Rouch´e-Fr¨obenius Theorem) Let us consider the system of
m linear equations with n unknowns in its matrix form A · x = b. Then:
- The above system is uniquely determined if and only if rk(A) = rk(A
∗ ) = n.
- The above system is underdetermined if and only if rk(A) = rk(A
∗ ) < n.
- The above system is inconsistent if and only if rk(A) 6 = rk(A
∗ ).
Remark 1.3.6 Taking into account the above result, only systems having associat-
ed matrices (i.e., coefficient and augmented matrices) whose size allows to find their
rank by the reader will be discussed and solved in this course.
1.3.2. Resolution of a System of Linear Equations
There exist several methods to solve linear systems. Let us remember two by
applying them to the following examples:
- Substitution method:
2 x + 3y = 4
−x + 2y = 3
y =
4 − 2 x
3
−x + 2y = 3
y =
4 − 2 x
3
−x + 2
4 − 2 x
3
y =
4 − 2 x
3
x = −
1
7
y =
10
7
x = −
1
7
- Elimination method:
2 x + 3y = 4
−x + 2y = 3
2 x + 3y = 4
− 2 x + 4y = 6
2 x + 3y = 4
7 y = 10
2 x + 3 ·
10
7
y =
10
7
x = −
1
7
y =
10
7
Definition 1.3.7 A linear system is said to be homogeneous if every constant
term equals zero, (that is, b = 0 = θ).
Remark 1.3.8 Every homogeneous linear system A · x = 0 is consistent, having at
least the so called trivial solution, that is, the zero vector 0 ∈ IR
n .
The set of all solutions of a homogeneous linear system is a vector subspace.
