Matrices and determinants

CHAPTER I
MATRICES AND DETERMINANTS
Department of Foundation Year,
Institute of Technology of Cambodia
2016–2017
CALCULUS II () ITC 1 / 56

Contents
1 Deﬁnitions
2 Reduced Row Echelon Form of Matrix
3 Operations and Properties
4 System of Linear Equations
5 Inverse Matrix
6 Rank and Nullity of a Matrix
7 Polynomial of a Matrix
8 Deﬁnitions
9 Properties of Determinants
10 Minor and Cofactor
11 Cramer’s rule

Contents
1 Deﬁnitions
5 Inverse Matrix
8 Deﬁnitions
11 Cramer’s rule

Matrices
Notation: Let K be the set of all real numbers or the set of all
complex numbers and aij be the elements of K.
Deﬁnition 1
A matrix is a rectangular array of numbers or symbols. It can be
written as





a11 a12 . . . a1m
a21 a22 . . . a2m
...
...
...
...
am1 am2 . . . amn





or





a11 a12 . . . a1m
a21 a22 . . . a2m
...
...
...
...
am1 am2 . . . amn





We denote this array by the single letter A or by A = (aij)m×n, and we
say that A has m rows and n columns, or that it is an m × n matrix.
We also say that A is a matrix of size m × n or of order m × n . The
number aij in the ith row and jth column is called the (i, j) entry.

Matrices
Note that the first subscript on aij always refers to the row and the
second subscript to the column.
Definition 2 (Zero matrix)
A zero matrix, denoted O, is a matrix whose all entries zero.
Definition 3 (Square matrix)
A square matrix is a matrix with the same number of rows as
columns. That is,
(aij)n =





a11 a12 . . . a1n
a21 a22 . . . a2n
...
...
...
...
an1 an2 . . . ann





The main diagonal of a square matrix is the list of entries
a11, a22, . . . , ann.

Matrices
Deﬁnition 4 (Upper triangular matrix)
A matrix TU is upper triangular if it is a square matrix whose
entries below the main diagonal are all zeros. That is,
TU =





a11 a1,2 . . . a1n
0 a22 . . . a2n
...
...
...
...
0 0 . . . ann





or TU =





a11 a1,2 . . . a1n
a22 . . . a2n
...
...
ann






Matrices
Deﬁnition 5 (Lower triangular matrix)
A matrix TL is lower triangular if it is a square matrix whose entries
above the main diagonal are all zeros. That is,
TL =






a11 0 . . . 0
a21 a22
... 0
...
...
...
...
an1 an2 . . . ann






or TL =





a11
a21 a22
...
...
...
an1 an2 . . . ann





Deﬁnition 6 (Triangular matrix)
A matrix T is triangular if it is upper triangular or lower triangular.

Matrices
Deﬁnition 7 (Diagonal matrix)
A diagonal matrix is a square matrix with all the entries which are
not on the main diagonal equal to zeros. So D = (aij) is diagonal if it
is n × n and aij = 0 for i = j. That is,
D =






a11 0 . . . 0
0 a22
... 0
...
...
...
...
0 0 . . . ann






or D =





a11
a22
...
ann





If aii = 1 for i = 1, 2, . . . , n, then the above matrix is called identity
matrix and denoted by In or I.

Contents
1 Deﬁnitions
5 Inverse Matrix
8 Deﬁnitions
11 Cramer’s rule

Row Echelon Form and Row Echelon Matrix
Deﬁnition 8
An m × n matrix is said to be in row echelon form (and will be
called a row echelon matrix) if it has the following three properties:
1 Every non-zero row begins with a leading one.
2 A leading one in a lower row is further to the right.
3 Zero rows are at the bottom of the matrix
Example 9
A =


1 2 4 0
0 1 5 2
0 0 1 4

 , B =


1 0 0 1
0 0 1 4
0 0 0 1

 , C =


1 0 0 0
0 0 0 1
0 0 0 0



Reduced Row Echelon Form and Reduced Row Echelon
Matrix
Deﬁnition 10
An m × n matrix is said to be in reduced row echelon form (and
will be called a reduced row echelon matrix) if it has the following
four properties:
1 Every non-zero row begins with a leading one.
2 A leading one in a lower row is further to the right.
3 Zero rows are at the bottom of the matrix.
4 Every column with a leading one has zeros elsewhere.

Elementary Row Operations
Deﬁnition 11 (Elementary Row Operations)
Ei,j: This is shorthand for the elementary operation of switching
the ith and jth rows of the matrix.
Ei(λ): This is shorthand for the elementary operation of
multiplying the ith row by the nonzero constant λ.
Ei,j(λ): This is shorthand for the elementary operation of adding
λ times the jth row to the ith row.

Elementary Matrices
Deﬁnition 12 (Elementary matrices)
An elementary matrix of size n is obtained by performing the
corresponding elementary row operation on the identity matrix In. We
denote the resulting matrix by the same symbol as the corresponding
row operation.
Example 13
Find the reduced row echelon form of the following matrices
A =


1 2 4
−2 3 5
0 −1 7

 , B =
3 −3 3
2 −1 4

Contents
1 Deﬁnitions
5 Inverse Matrix
8 Deﬁnitions
11 Cramer’s rule

Transpose
Definition 14
Let A = (aij)m×n. Then the transpose of A is the n × m matrix At
obtained by interchanging the rows and columns of A. That is,
At
=





a11 a12 . . . a1m
a21 a22 . . . a2m
...
...
...
...
am1 am2 . . . amn





t
=





a11 a21 . . . am1
a12 a22 . . . am2
...
...
...
...
a1n a2n . . . amn





Definition 15
The matrix A is said to be symmetric if At = A.
Definition 16 (Skew-symmetric matrix)
A matrix A is said to be skew-symmetric matrix if At = −A.

Operation on matrices
Deﬁnition 17 (Equality)
Two matrices are equal if they have the same size and if corresponding
entries are equal. That is, if A = (aij)m×n and B = (bij)m×n, then
A = B ⇐⇒ aij = bij, 1 ≤ i ≤ m, 1 ≤ j ≤ n.

Definition 18
Let A = (aij)m×n and B = (bij)m×n and let λ ∈ K, then
A + B = (aij + bij)m×n (addition)
λA = (λaij)m×n (scalar multiplication)
Definition 19
Let A = (aij)m×n and B = (bij)n×p, then the product of matrices A
and B is defined by
AB = C = (cij)m×p
where
cij =
n
k=1
aikbkj = ai1b1j + ai2b2j + · · · + ainbnj
for 1 ≤ i ≤ m and 1 ≤ j ≤ p.

Theorem 1
Let M be the set of all m × n matrices. For A, B, C ∈ M and
λ1, λ2 ∈ K, we have
1 (Closure law) A + B ∈ M
2 (Associative law) (A + B) + C = A + (B + C)
3 (Commutative law) A + B = B + A
4 (Identity law) A + O = A
5 (Inverse law) A + (−A) = O
6 (Closure law) λ1A ∈ M
7 (Associative law) λ1(A + B) = λ1A + λ1B
8 (Distributive law) (λ1 + λ2)A = λ1A + λ2A
9 (Distributive law) (λ1λ2)A = λ1(λ2A)
10 (Monoidal law) 1A = A

Theorem 2
Let A, B, C be matrices of the appropriate sizes so that the following
multiplications make sense, I a suitably sized identity matrix, and λ a
scalar. Then
AB is m × n matrix
(AB)C = A(BC)
IA = A and AI = A
A(B + C) = AB + AC
(B + C)A = BA + CA
λ(AB) = (λA)B = A(λB)
Ei,jEi,j = I, Ei,j(λ)Ei,j(−λ) = I and Ei(λ)Ei(1/λ) = I
Deﬁnition 20
The matrix A is said to be orthogonal if AtA = I.

Theorem 3 (Laws of Matrix Transpose)
Let A and B be matrices of the appropriate sizes so that the following
operations make sense, and λ a scalar, then
(A + B)t = At + Bt
(At)t = A
(λA)t = λAt
(AB)t = BtAt
(Ei,j)t = Ei,j, (Ei,j(λ))t = Ej,i(λ), (Ei(λ))t = Ei(λ)

Trace of a Matrice
Deﬁnition 21
Let A = (aij)n. The trace of the matrix A is deﬁned by
tr(A) =
n
i=1
aii
Theorem 4
Let A and B be two square matrices, and λ be a scalar. Then
tr(A + B) = tr(A) + tr(B)
tr(λA) = λtr(A)
tr(AB) = tr(BA).

Contents
1 Deﬁnitions
5 Inverse Matrix
8 Deﬁnitions
11 Cramer’s rule

System of Linear Equations
Deﬁnition 22
A system of m linear equations in the n unknowns x1, x2 . . . , xn is a list
of m equations of the form
a11x1 + a12x2 + · · · + a1nxn = b1
a11x1 + a12x2 + · · · + a1nxn = b2 (S)
...
...
...
am1x1 + am2x2 + · · · + amnxn = bm
The system of equations (S) is called homogeous if bi = 0 for
i = 1, 2, . . . , m. Otherwise, the system is said to be nonhomogeneous.

Deﬁnition 23
A solution vector for the general linear system given by Equation (S)
is an n × 1 matrix
X =





s1
s2
...
sn





such that the resulting equations are satisﬁed for these choices of the
variables. The set of all such solutions is called the solution set of the
linear system, and two linear systems are said to be equivalent if they
have the same solution sets.
The nonhomogeneous system of equation is called consistent if it has
solutions.

Deﬁnition 24 (Elementary Operations)
There are three elementary operations that will transform a linear
system into another equivalent system:
1 Interchanging two equations
2 Multiplying an equation through by a nonzero number
3 Adding to one equation a multiple of some other equation
Objective: The goal is to transform the set of equations into a simple
form so that the solution is obvious. A practical procedure is suggested
by the observation that a linear system, whose coeﬃcient matrix is
either triangular or diagonal, is easy to solve

Let
A =





a11 a12 . . . a1n
a21 a22 . . . a2n
...
...
...
...
an1 an2 . . . ann





X =





x1
x2
...
xn





and b =





b1
b2
...
bm





Matrix A is called coeﬃcient matrix.
The matrix
(A|b) =





a11 a12 . . . a1n b1
a21 a22 . . . a2n b3
...
... . . .
...
...
am1 am2 . . . amn bm





is called augmented matrix.

So the above system of linear equations (S) can be written in matrix
equation
AX = b.
Deﬁnition 25 (Row Equivalent)
An m × n matrix A is said to be row equivalent to an m × n matrix B
if B can be obtained by applying a ﬁnite sequence of elementary row
operations to A.

Gauss Elimination Method
The Gauss Elimination procedure for solving the linear system
AX = b is as follows.
Form the augmented matrix (A|b).
Transform the augmented matrix to row echelon form by using
elementary row operations
Use back substitution to obtain the solution
Example 26
Solve a system of linear equations
a)



x1 + x2 + x3 = 2
2x1 + 3x2 + x3 = 3
x1 − x2 − 2x3 = −6
b)



4x1 + 8x2 − 12x3 = 44
3x1 + 6x2 − 8x3 = 32
−2x1 − x2 = −7

Gauss-Jordan Elimination Method
The Gauss-Jordan Elimination procedure for solving the linear
system AX = b is as follows.
Form the augmented matrix (A|b).
Transform the augmented matrix to reduced row echelon form by
using elementary row operations
The linear system that corresponds to the matrix in reduced row
echelon form that has been obtained in Step 2 has exactly the
same solutions as the given linear system. For each nonzero row of
the matrix in reduced row echelon form, solve the corresponding
equation for the unknown that corresponds to the leading entry of
the row.

Contents
1 Deﬁnitions
5 Inverse Matrix
8 Deﬁnitions
11 Cramer’s rule

Inverse Matrix
Deﬁnition 27 (Inverse Matrix)
The square matrix A is invertible if there is a matrix B such that
AB = BA = I, where I is the identity matrix. The matrix B is called
the inverse of A and is denoted by A−1.
Theorem 5
If A is an invertible matrix, then the matrix A−1 is unique.
Theorem 6
Let A and B two square matrices. Then
(AB)−1 = B−1A−1
(At)−1 = (A−1)t

Inverse Matrix
Theorem 7
Every nonzero m × n matrix is row equivalent to a unique matrix in
reduced row echelon form by a ﬁnite sequence of elementary row
operations.
Theorem 8
Let the matrix B be obtained from the matrix A by performing a ﬁnite
sequence of elementary row operations on A. Then B and A have the
same reduced row echelon form.

Contents
1 Deﬁnitions
5 Inverse Matrix
8 Deﬁnitions
11 Cramer’s rule

Rank and Nullity of a Matrix
Deﬁnition 28 (Rank of a matrix)
The rank of a matrix A is the number of nonzero rows of the
reduced row echelon form of A. This number is written as rank(A).
Deﬁnition 29 (Nullity of a matrix)
The nullity of a matrix A is the number of columns of the reduced row
echelon form of A that do not contain a leading entry. This number is
written as null(A).
Theorem 9
Let A be and m × n matrix. Then
0 ≤ rank(A) ≤ min{m, n}
rank(A) + null(A) = n.

Rank and nullity of a Matrix
Theorem 10
The general linear system (S) with m × n coefficient matrix A, in right
hand side vector b and augmented matrix (A|b) is consistent if and
only if rank(A) = rank(A|b), in which case either
1 rank(A) = n, in which case the system has a unique solution, or
2 rank(A) < n, in which case the system has infinitely many
solutions.
Theorem 11
If a consistent linear system of equations has more unknowns than
equations, then the system has infinitely many solutions.

Rank and nullity of a Matrix
Theorem 12
If A is an n × n matrix then the following statements are equivalent.
The matrix A is invertible.
There is a square matrix B such that AB = I.
The linear system AX = b has a unique solution for every right
hand side vector b.
The linear system AX = 0 has only the trivial solution.
rank(A) = n
The reduced row echelon form of A is I .
The matrix A is a product of elementary matrices.

Contents
1 Deﬁnitions
5 Inverse Matrix
8 Deﬁnitions
11 Cramer’s rule

Polynomial of a Matrix
Let A be a square matrix of order n. We form the power of matrix A
as follow:
A0
= I, A1
= A, A2
= AA, A3
= A2
A, . . . , An
= An−1
A
Let a0, a1, . . . , ak be scalars and
p(x) = a0 + a1x + a2x2
+ · · · + akxk
be a polynomial of degree k. We deﬁne a new matrix p(A) by
p(A) = a0I + a1A + a2A2
+ · · · + akAk
.
If a square matrix A such that p(A) = 0, then the matrix A is called a
zero of the polynomial p(x).

Contents
1 Deﬁnitions
5 Inverse Matrix
8 Deﬁnitions
11 Cramer’s rule

Permutations
Deﬁnition 30 (Permutations)
Let S = {1, 2, . . . , n} be the set of integers from 1 to n, arranged in
ascending order. A one-to-one mapping σ from S onto S is called a
permutation of S. We denote such a permutation σ by
σ =
1 2 . . . n
j1 j2 . . . jn
or σ = j1j2 . . . jn where ji = σ(i).
Observe that, since σ is one-to-one and onto, then the sequence
j1, j2, . . . , jn is simply a rearrangement of the numbers 1, 2, . . . , n.

Permutations
The set of all permutations of S is denoted by Sn. That is,
Sn = {j1j2 . . . jn : j1, j2, . . . , jn ∈ {1, 2, . . . , n} and ji = jk if i = k}.
We have the following:
#Sn = n! (the number of elements of Sn).
If σ ∈ Sn, then the inverse σ−1 ∈ Sn.
If σ, τ ∈ Sn, then the composition mapping σ ◦ τ ∈ Sn.
The identity permutation of S is denoted by where = 12 . . . n
and for σ ∈ Sn : σ ◦ σ−1 = σ−1 ◦ σ = .

Permutations
Definition 31
A permutation σ = j1j2 . . . jn of Sn is said to have an inversion if
a larger integer jr precedes a smaller one js.
A permutation is called even or odd according to whether the
total number of inversions in it is even or odd.
Definition 32
The signature of a permutation σ, denoted sgn(σ), is defined as
sgn(σ) =
1, if σis even
−1, if σis odd.

Permutations
Theorem 13
Let σ1, σ2 ∈ Sn. Then,
sgn(σ1 ◦ σ2) = sgn(σ1) ◦ sgn(σ2)
sgn(σ−1
1 ) = sgn(σ1)

Determinants
Deﬁnition 33
Let A = (aij)n be a square matrix of order n. The determinant of A
(written det(A) or |A| ) is deﬁned by:
|A| =
a11 a12 . . . a1n
a21 a22 . . . a2n
...
...
...
...
an1 an2 . . . ann
=
σ∈Sn
sgn(σ)a1σ(1)a2σ(2) . . . anσ(n)
=
σ∈Sn
sgn(σ)a1j1 a2j2 . . . anjn .

Contents
1 Deﬁnitions
5 Inverse Matrix
8 Deﬁnitions
11 Cramer’s rule

Properties of Determinants
Theorem 14
The determinants of a matrix A and its transpose At are equal, that is
|A| = |At|.
Theorem 15
If B is a matrix resulting from matrix A:
by multiplying a row of A by a scalar λ; that is B = Ei(λ), then
|B| = λ|A|.
by interchanging two rows of A; that is B = EijA, then
|B| = −|A|.
by adding to row i of A a constant λ times row j of A with i = j;
that is B = Eij(λ)A, then |B| = |A|.

Properties of Determinants
Theorem 16
Let A be a square matrix.
If a row(column) of A consists entirely of zeros, then |A| = 0.
If two rows (columns) of A are equal, then |A| = 0.
If A is triangular, then |A| = a11a22 . . . ann.
Theorem 17
The determinant of a product of two matrices is the product of their
determinants; that is
|AB| = |BA|.

Determinant
Example 34
Given matrix
A =


a11 a12 a13
a21 a22 a23
a31 a32 a33

 .
and |A| = 7
Compute the following determinants
a).|2A| b).|A−1
| c).|(3A)−1
| d).
a11 a31 a21
a12 a32 a22
a13 a33 a23

Contents
1 Deﬁnitions
5 Inverse Matrix
8 Deﬁnitions
11 Cramer’s rule

Minor and Cofactor
Deﬁnition 35
Let A = (aij)n and let Mij be the (n − 1) × (n − 1) sub matrix of A,
obtained by deleting ith row and jth column of A. The determinant
|Mij| is called the minor of aij of A. The cofactor Aij of aij is
deﬁned as Aij = (−1)i+j|Mij|.
Theorem 18 (Laplace expansion)
Let A = (aij)n. Then for each 1 ≤ i ≤ n,
|A| =
n
k=1
aikAik = ai1Ai1 + ai2Ai2 + · · · + ainAin
and for each 1 ≤ j ≤ n
|A| =
n
k=1
akjAkj = a1jA1j + a2jA2j + · · · + anjAnj

Minor and Cofactor
Theorem 19
Let A = (aij)n. Then
ai1Ak1 + ai2Ak2 + · · · + ainAkn = 0 for i = k
a1jA1k + a2jA2k + · · · + anjAnk = 0 for j = k
Example 36
Find the determinant of the following matrix using Laplace expansion
A =


1 2 3
−1 4 2
3 −1 0

 , B =


1 2 −1
3 0 1
4 2 1

 .

Minor and Cofactor
Deﬁnition 37 (Adjoint)
Let A = (aij)n be a square matrix of order n. The n × n matrix
adj(A), called the adjoint of A, is the matrix whose (i, j) entry is the
cofactor Aji of aji. Thus
adj(A) =





A11 A12 . . . A1n
A21 A22 . . . A2n
...
...
...
...
An1 An2 . . . Ann





t

Minor and Cofactor
Theorem 20
Let A be a square matrix. Then,
A.adj(A) = (adj(A))A = |A|I.
If |A| = 0, then A−1
=
1
|A|
(adj(A)).
Example 38
Find inverse matrix of the below matrix
A =


1 2 −4
0 2 3
1 1 −1

 .

Contents
1 Deﬁnitions
5 Inverse Matrix
8 Deﬁnitions
11 Cramer’s rule

Cramer’s rule
Considers a system of n linear equations in the n unknowns:
a11x1 + a12x2 + · · · + a1nxn = b1
a21x1 + a22x2 + · · · + a2nxn = b2
...
an1x1 + an2x2 + · · · + annxn = bn
This system is equivalent to the matrix form
AX = b
where
A =





a11 a12 . . . a1n
a21 a22 . . . a2n
...
...
...
...
an1 an2 . . . ann





, b =





b1
b2
...
bn






Cramer’s rule
We denote ∆ the determinant of the coeﬃcient matrix A = (aij)n, that
is ∆ = |A| and ∆xi the determinant of the matrix obtained by
replacing the ith column of A by the column of b.
Theorem 21
The above system has a unique solution if and only if ∆ = 0. In this
case, the unique solution is given by
x1 =
∆x1
∆
, x2 =
∆x2
∆
, . . . , xn =
∆xn
∆
.

Cramer’s rule
Theorem 22
Let A be a square matrix. The following statement are equivalent.
A is invertible.
AX = 0 has only the trivial solution.
The determinant of A is not zero.
Theorem 23
The homogeneous system AX = 0 has a nontrivial solution if and only
if ∆ = |A| = 0.

Matrices and determinants

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Matrices and determinants