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Mathematics 1
Applied Informatics
Štefan BEREŽNÝ
7th lecture
Contents
LINEAR ALGEBRA
• Matrices
• Determinants
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Matrix
Definition:
A rectangular array of mn real numbers written in m rows and n
columns is called a matrix of the type m  n (read: type m by n or
shortly an m by n matrix). The numbers which are contained in
the matrix are called its entries or its elements. Matrices are
usually denoted by capital letters and their entries are denoted
by the same small letters with two indices. The indices are
related to the position of the entry aij (i-th row and j-th column in
matrix A).
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Matrix
Specially type of matrices:
(a) Upper triangular matrix: If all elements under
the main diagonal are equal to zero, then
matrix A is called the upper triangular matrix.
(b) Lower triangular matrix: If all elements above
the main diagonal are equal to zero, then
matrix A is called the lower triangular matrix.
(c) Zero matrix: A matrix whose all elements are
equal to zero is called zero matrix.
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Matrix
Specially type of matrices:
(d) Transposed matrix: The n  m matrix B = bij
whose elements satisfy bij = aji for i = 1, 2, ..., m
and j = 1, 2, ..., n is called a transposed
matrix to matrix A. It is denoted by AT. In other
words: the transposed matrix AT to matrix A can
be obtained by turning A over the main
diagonal.
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Matrix
Specially type of matrices:
(e) Square matrix: A matrix with the same
number of rows as columns is said to be
a square matrix.
(f) Diagonal matrix: The square n  n matrix that
is upper and lower triangular matrix together is
called diagonal matrix. It is denoted by diag(A)
= a11, a22, a33, …, ann.
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Matrix
Specially type of matrices:
(g) Identity matrix: The diagonal matrix A whose
elements equal 1 is called the identity matrix.
It is denoted by E or I.
(h) Symmetric matrix: The square matrix A is
symmetric if satisfy: A = AT.
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Matrix
Two matrices are identical if they are of the same
type and if they have the same entries at
corresponding position.
Suppose that A = aij is an m  n matrix. The
entries a11, a22, a33, …, akk (where k = minm, n)
form a so called main diagonal in matrix A.
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Matrix operations
Addition of matrices:
If matrices A = aij and B = bij are both m  n
then their sum is the m  n matrix C = cij with
elements cij = aij + bij for i = 1, 2, …, m and j = 1, 2,
…, n. We use the notation C = A + B.
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Matrix operations
Multiplication of matrices by real numbers:
If A = aij is an m  n matrix and   R, then the
product of the number  and matrix A is the
matrix C = cij of the same type m  n with
elements cij = aij for i = 1, 2, …, m and j = 1, 2,
…, n. We say: matrix C is -multiple of matrix A.
We use the notation C = A or C = A.
Matrices of the same type can also be subtracted.
The difference of matrices A and B is the matrix
C = A + (1)B = A  B.
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Matrix operations
Multiplication of matrices:
If A = aij is an m  k matrix and B = bij is k  n
then the product of the matrices A and B is the
m  n matrix C = cij whose elements satisfy:
k
cij   ail  blj
l 1
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Matrix operations
for i = 1, 2, …, m and j = 1, 2, …, n.
We write: C = AB.
You can observe that the element cij in matrix C is
the scalar product of i-th row of matrix A with j-th
column of matrix B.
Holds:
Štefan BEREŽNÝ
Multiplication of matrices is not
commutative!
MATHEMATICS 1
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Matrix operations
Rules for operations with matrices:
(1)
(2)
(3)
(4)
(5)
A + B = B + A,
(A + B) + C = A + (B + C),
(A  B)  C = A  (B  C),
(A + B) = A + B,
( + )A = A + A,
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Matrix operations
Definition:
The maximum number of linearly independent rows (or
linearly independent columns) is called the rank of matrix
A. We denote it rank(A) or r(A). Rows and columns are
taken as arithmetic vectors.
Theorem:
Let A be an m  n upper triangular matrix and let all
elements on the main diagonal be different from zero. Then
the rank of matrix A is equal to the minimum of the
numbers m and n.
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Matrix operations
Elementary row and column operations:
We can transform the non-upper triangular matrix
to an upper triangular matrix using so called
elementary row and column operations, which do
not change the rank of the matrix. We shall use the
following elementary row operations:
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Matrix operations
(a)
(b)
(c)
(d)
change of order of rows,
multiplication of some row by a nonzero real
number,
addition to some row of a linear combination
of the other rows,
omission of a row which is a linear
combination of the other rows.
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Matrix operations
All the operations can also be performed with
columns.
The procedure of transformation of an arbitrary
matrix to an upper triangular matrix (all of whose
elements on the main diagonal are different from
zero) by means of the elementary row and column
operations is called the Gauss algorithm.
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Determinant
Definition:
Let A be a square matrix. The determinant of matrix A is
the number which is denoted by det(A) and which is
assigned to matrix A in accordance with these rules:
(a)
If A = a is a 1  1 square matrix then det(A) = a.
(b)
If A = aij is an n  n square matrix (for n  1) then
we choose an arbitrary i-th row of matrix A and we put:
det(A) = ai1Ai1 + ai2Ai2 + ai3Ai3 + … + ain1Ain1 + ainAin
where Aij is a so called co-factor of element aij.
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Determinant
The co-factor is equal to (1)i+jdet(Aij) where det(Aij) is
the determinant of the (n1)  (n1) square matrix
which arises from A by omission the i-th row and the jth column. det(Aij)is called the minor, which is the
abbreviation for “minor determinant”. The sum ai1Ai1 +
ai2Ai2 + ai3Ai3 + … + ain1Ain1 + ainAin is called the
expansion of the determinant according to the i-th
row. The expansion of the determinant according to
the j-th column is:
det(A) = a1jA1j + a2jA2j + a3jA3j + … + an1jAn1j + anjAnj.
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Determinant
Saruss' rule:
The determinant of a 2  2 and 3  3 matrix can
also be, apart from the expansion according to
some row or column, computed by the so called
“Saruss' rule”:
det(A) = a11a22 + a12a21
and
det(A) = a11a22a33 + a12a23a31 + a13a21a32 
(a13a22a31 + a11a23a32 + a12a21a33).
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Determinant
Important facts about determinants:
(1) If all elements in some row or column of matrix
A are zero then det(A) = 0.
(2) Interchanging two rows or columns changes
the sign of the determinant.
(3) If two rows or columns are identical the
determinant is zero.
(4) If we multiply some row or column of matrix A
by a real number  then the determinant of the
new matrix is equal to det(A).
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Determinant
Important facts about determinants:
(5) If any row respectively column of matrix A is a
multiple of another row respectively column of A,
the determinant of A is zero.
(6) If any row respectively column of matrix A is a
linear combination of the other rows respectively
columns of matrix A, the determinant is zero.
(7) det(A) = det(AT).
(8) If A and B are n  n square matrices then det(A  B)
= det(A)  det(B).
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Determinant
Definition:
An n  n square matrix which has the maximum
possible rank is called a regular matrix. (i.e.
rank(A) = n)
Definition:
Suppose that A is an n  n square matrix and E is
the n  n identity matrix. An n  n square matrix A1
is called the inverse matrix to matrix A if:
A  A1 = A1  A = E.
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Determinant
Theorem:
Let A be a square matrix. Then the following
statements are equivalent:
(1) A is regular.
(2) det(A)  0.
(3) The inverse matrix A1 exists.
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Determinant
Theorem:
If A and B are n  n regular matrices then matrix
A  B is also regular matrix. Moreover, it holds:
(A  B)1 = B1  A1.
Theorem:
If matrix A is regular then matrix A1 is also regular.
Moreover, it holds:
(1)
(2)
(A1)1 = A,
A  A1 = A1  A = E.
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Determinant
Theorem:
(Uniqueness of the inverse matrix)
If a square matrix A has an inverse matrix then the
inverse matrix A1 is unique.
Let A is an n  n square regular matrix. For inverse
matrix A1 to matrix A holds:
1
A 
 Adj( A)
det( A)
1
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Determinant
The adjoint matrix Adj(A) of matrix A is the
transpose matrix of the matrix of co-factors of the
elements aij in matrix A.
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Thank you for your attention.
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