Matrix

Definition: A matrix is a rectangular array of numbers enclosed by a pair of

brackets and is denoted by a capital letters A, B etc.

𝑎!! 𝑎!" … 𝑎#!

𝑎"! 𝑎"" … 𝑎#"
𝐴=# … … … … &= [aij]
𝑎$! 𝑎$" … 𝑎$#

Numbers aij (i =1, 2, …m, j=1,2, …n) are called elements. First subscript
indicates the row; second subscript indicates the column. The matrix consists
of mn elements.

Elements a11, a22, a33, ... ann called diagonal elements.

Number of rows(m) by number of columns(n) of a matrix is called order of the

matrix and is written as m x n.

Types of matrices

Square matrices: If the number of rows of a matrix A is equal to the number

of columns i.e., m = n, then the matrix A is called square matrix of order n x n.

𝑎!! 𝑎!" … 𝑎#!

𝑎"! 𝑎"" … 𝑎#"
𝐴=# … … … … &
𝑎$! 𝑎$" … 𝑎$#

Elements a11, a22, a33, ..., amn called diagonal elements.

Column matrices: A matrix with only one column is called Column Matrix.
i,e,
1
𝐴 = '0+
8

Matrix A is a column matrix of Order 3x1

Row matrices: A matrix with only one row is called row Matrix. i,e,
𝐴 = [1 2 3]

Matrix A is a column matrix of Order 1x3

Equal matrices: Two matrices A= [aij] & B= [bij] are said to be equal (A = B)
if and only if each element of A is equal to the corresponding element of B, i.e.,
aij = bij for 1 ≤ i ≤ m, 1 ≤ j ≤ n.
if A = B, it implies aij = bij for 1 ≤ i ≤ m, 1 ≤ j ≤ n;
if aij = bij for 1 ≤ i ≤ m, 1 ≤ j ≤ n, it implies A = B.

1 0 𝑎 𝑏
Example: A= 1 4 B= 1 4
−4 2 𝑐 𝑑

if A = B, then a = 1, b = 0, c = -4 and d = 2.

Zero matrices: Every element of a matrix is zero, it is called a zero matrix, i.e.,

0 0 … 0
0 0 … 0
𝐴=# &
… … … …
0 0 … 0
Upper triangular matrix: A square matrix whose elements aij = 0, for i > j is
called upper triangular matrix, Example,

𝑎!! 𝑎!" … 𝑎#!

0 𝑎"" … 𝑎#"
𝐴=#… … … … &
0 0 … 𝑎$#
Lower triangular matrix: A square matrix whose elements aij = 0, for i < j is
called lower triangular matrix, Example,

𝑎!! 0 … 0
𝑎 𝑎"" … 0
𝐴 = # "! &
… … … …
𝑎$! 𝑎$" … 𝑎$#
Diagonal matrix: A square matrix whose elements aij = 0, for i ¹ j is called
diagonal matrix. Example ,
𝑎!! 0 … 0
0 𝑎"" … 0
𝐴=# &
… … … …
0 0 … 𝑎$#
and is denoted by D= diag[a11, a22,….. amn]

Identity matrix or Unit Matrix: A square matrix whose elements aij = 0, for
i ¹ j and aij = 1, for i = j is called identity matrix or unit matrix and is denoted
by I.
1 0 0
Examples of identity matrix I3 = '0 1 0+
0 0 1

Properties of identity matrix: AI = IA = A

Transpose matrix: The matrix obtained by interchanging the rows and
columns of a matrix A is called the transpose of A (write AT or A’).

For a matrix A = [aij], its transpose AT = [bij], where bij = aji.

1 2 3 1 4 7
Example: A= '4 5 6+, Transpose of A is A = '2 5 8+T

7 8 9 3 6 9

Properties
▪ (AT)T = A and (lA)T =lAT
▪ (A + B)T = AT + BT
▪ (A - B)T = AT- BT
▪ (AB)T = BT AT

Symmetric Matrix: A square matrix A whose elements aji = aij for all i and j
is called symmetric matrix. Example,

1 2 3
𝐴 = '2 4 −5+ is a symmetric matrix
3 −5 6

Properties: AT = A

Skew Symmetric Matrix: A square matrix A, whose elements aji = -aij for all
i ¹j and aij = 0 for i = j is called skew symmetric matrix. Example,

0 2 −1
𝐴 = '−2 0 2 + is a skew symmetric matrix
1 −2 0

Properties: AT = - A
Determinant of a square matrix
Determinant whose elements are exactly the same as those of a square matrix
A, is called the determinant of the matrix A and is denoted by |𝐴|.

Singular matrix: In the determinant of a square matrix A, if |𝐴| = 0, then the

matrix is called singular matrix.
1 2
Thus A= 1 4 is a singular matrix,
3 6
1 2
since |𝐴|= = ==0
3 6

Non-singular matrix: In the determinant of a square matrix A, if |𝐴| ≠ 0, then

the matrix is called non-singular matrix.
1 2
Thus A= 1 4 is a singular matrix,
2 6
1 2
since |𝐴|= = ==2≠0
2 6

Adjoint of a square matrix: The adjoint of a square matrix A is the transpose

of the matrix formed by the cofactors of the elements of the determinant of the
matrix A and is denoted by adjA.

Properties of adjoint matrix:

1. A(adjA) = (adjA)A = |𝐴| I
2. adjAB = (adjB)(adjA)
3. (adjA)T = adj(AT)

Inverse Matrix: If two non-singular (i,e, |𝐴|¹ 0) square matrices A and B such
that AB = BA = I, then B is called the inverse of A and is denoted by the symbol
A-1 and is defined by
%&'(
A-1= |(|
Properties:
1. (A-1)T = (AT)-1
2. (AB)-1 = B-1A-1
Minors of elements: If A is a square matrix, then the minor of the element in
the i-th row and j-th column i,e, aij is the determinant of the submatrix formed
by deleting the i-th row and j-th column.

Cofactor of elements: The cofactor of an element aij is obtained by multiplying

the minor of element aij by (-1)i+j and is denoted by Aij.

Orthogonal Matrix: A Matrix is an Orthogonal Matrix when the product of a

matrix A and its transpose AT gives an identity value.

AAT = I
or, A = A-1

Rank
Every matrix has a rank. A non-zero matrix is said to have rank r if at least one
of its minors of order r is nonzero, while every minor of order (r+1), if any, is
zero. A zero matrix is said to have rank 0. The statement the rank of A is r then
it is denoted ρA = r

Equivalence matrices: Two matrices are called equivalent if one can be

obtained from the other by a sequence of elementary transformations. Two
equivalent matrices A and B are symbolically written as A ~ B.

Canonical matrix: A non-zero matrix A of rank r is row equivalent to a unique

matrix C, called a canonical matrix of A which is obtained from A

Characteristic matrix: For a given square matrix A, A − λI matrix is called

characteristic matrix, where λ is scalar and I is the unit matrix.

Characteristic polynomial matrix: For a given square matrix A, the

determinant |𝐴 − 𝜆𝐼 | is called characteristic polynomial of matrix A.

Characteristic equation: The equation |𝐴 − 𝜆𝐼| = 0 is called characteristic

equation of matrix A.
Properties of eigen values:
i) Any square matrix A and its transpose AT have the same eigen values.
ii) The product of eigen values of a square matrix A is equal to the determinant
of A.
iii) If λ1, λ2, λ3, …λn are the eigen values of a matrix A, then
(a) kλ1, kλ2, kλ3, …kλn, are the eigen values of kA,
! ! ! !
(b) , , , … are the eigen values of A−1.
*! *" *+ *,

Cayley-Hamilton Theorem:

Statement: Every square matrix satisfies its characteristic equation.

If |𝐴 − 𝜆𝐼|= a0 + a1λ + a2λ2 + …anλn = 0 be the characteristic equation of an
n × n matrix A, then a0I + a1A + a2A2 + − − −anAn = 0
Properties of matrices:
Matrices A, B and C are conformable,
1. A + B = B + A (commutative law)
2. A + (B +C) = (A + B) +C (associative law)
3. l(A + B) = lA + lB, where l is a scalar. (distributive law)
4. A(B + C) = AB + AC
5. (A + B)C = AC + BC
6. A(BC) = (AB) C
7. AB ¹ BA in general
8. AB = 0 NOT necessarily imply A = 0 or B = 0
9. AB = AC NOT necessarily imply B = C
10. (AB)-1 = B-1A-1
11. (A-1)T = (A-T)-1
12. (AT)T = A and (lA)T = lAT
13. (A + B)T = AT + BT
14. (AB)T = BTAT
15. |𝐴- | = |𝐴|
16. |𝐴𝐵| = |𝐴||𝐵|