The document defines and provides examples of different types of matrices such as square, diagonal, identity, and zero matrices. It also discusses matrix operations like addition, subtraction, scalar multiplication, and matrix multiplication. Key properties of these operations such as commutativity, associativity, and invertibility are covered. Matrix transpose and elementary row operations are also introduced.

2. A matrix is an ordered rectangular
array of numbers or functions.

3.  A =
4 2 6 6
−1 0 2 7
 B =
1 9
−8 1
3 5
 C =
1 3 6
8 11 −1
5 −2 7
0 4 0

4. The numbers or functions are called
the elements or the entries of the
matrix.

5.  The number of rows and columns of a matrix is
defined as the order of the matrix.
 Matrix having m rows and n columns is called
(m × n) matrix.
 Order of the matrix = m x n
 Read as m by n

6.  A =
4 2 6 6
−1 0 2 7
Order of A = 2 x 4
 B =
1 9
−8 1
3 5
Order of B = 3 x 2
 C =
1 3 6
8 11 −1
5 −2 7
0 4 0
Order of C = 4 x 3
 D = 5 9 2
Order of D = 1 x 3
E =
7
4
Order of E = 2 x 1

7.  By the Order of the Matrix , number of
elements can be calculated by the product of
the row and column.
 B =
1 9
−8 1
3 5
Order of B = 3 x 2
No of elements = 6

8.  If the No. of elements is given, all possible
products of its factors is the order.
 If a Matrix has 12 elements, what are the
possible orders it can have?
 (1 x 12), (2 x 6), (3 x 4), (4 x 3), (6 x 2), (12 x 1)

9. A =
𝑎11 𝑎12 𝑎13 … … … … . 𝑎1𝑗 … … . … . 𝑎1𝑛
𝑎21 𝑎22 𝑎23 … … … . … 𝑎2𝑗 … … . … . 𝑎2𝑛
⋮ ⋮ ⋮ ⋮ ⋮
⋮ ⋮ ⋮ ⋮ ⋮
𝑎𝑖1 𝑎𝑖2 𝑎𝑖3 … . … . … . 𝑎𝑖𝑗 … … . … . 𝑎𝑖𝑛
⋮ ⋮ ⋮ ⋮ ⋮
⋮ ⋮ ⋮ ⋮ ⋮
𝑎 𝑚1 𝑎 𝑚2 𝑎 𝑚3 … … . … . 𝑎 𝑚𝑗 … … . … 𝑎 𝑚𝑛
ie., A = [aij]mxn where 1 ≤ i ≤ m , 1 ≤ j ≤ n
 Matrices are named by Capital alphabets
 Elements with small alphabets
 R represents the Rows
 C represents the Columns

10.  B =
𝑏11 𝑏12
𝑏21 𝑏22
𝑏31 𝑏32
𝑏41 𝑏42
𝑏51 𝑏52

11.  Costruct a 2x3 matrix where aij = 2i + j
 a11 = 2(1) + 1 = 2 + 1 = 3
 a12 = 2(1) + 2 = 2 + 2 = 4
 a13 = 2(1) + 3 = 2 + 3 = 5
 a21 = 2(2) + 1 = 4 + 1 = 5
 a22 = 2(2) + 2 = 4 + 2 = 6
 a23 = 2(2) + 3 = 4 + 3 = 7

3 4 5
5 6 7

12. 
3 5
6 8
11 13

13. A matrix is said to be a row matrix if it has only
one row
In general, A = (𝑎𝑖𝑗)1 x n & order = 1 x n
eg : (6 5 4 7 9 )

14.  A matrix is said to be a column matrix if it has
only one column.
In general, A = (𝑎𝑖𝑗)m x 1 & order = m x 1
eg :
6
8
6

15. A matrix in which the number of rows are
equal to the number of columns
In general, A = (𝑎𝑖𝑗)m x m & order = m
Eg:
5 7
2 5
,
−2 0 7
3 8 9
1 6 4

16.  A square matrix is said to be a diagonal matrix
if all the elements except those in the leading
diagonal are zero.
 ie., B = [bij ]mxn is a diagonal matrix if
bij = 0 when i ≠ j
 Eg: 3 0 0
0 7 0
0 0 −1

17.  A diagonal matrix is said to be a scalar matrix if
its diagonal elements are equal
 ie., B = [bij ]mxn is a scalar matrix if
bij = 0 when i ≠ j
bij = k when i = j Where k is some constant
 Eg: 2 0 0
0 2 0
0 0 2

18.  A square matrix in which the diagonal
elements are all 1 and rest are all zero is called
an identity matrix or unit matrix.
 B = [bij ]mxn is a unit matrix if
bij = 0 when i ≠ j
bij = 1 when i = j
 Eg:
1 0 0
0 1 0
0 0 1

19.  A matrix is said to be zero matrix or null
matrix or void matrix if all its elements are
zero.
 Zero matrix can be of any order.
 Eg:
0 0
0 0

20.  Row Matrix
 Column Matrix
 Square Matrix
 Diagonal Matrix
 Scalar Matrix
 Identity Matrix (or) Unit Matrix
 Zero Matrix (or) Null Matrix

21.  Two matrices A = [aij] and B = [bij] are equal if
 (i) They are of the same order .
(ii) Corresponding elements are equal
ie., aij = bij for all i and j.
 Eg: A =
4 2 6
−1 0 7
, B =
4 2 6
−1 0 7
A = B

22. 
𝑥 + 𝑦 2
𝑧 − 5 𝑦
=
8 2
9 7
 By equating we get, 𝑥 + 𝑦 = 8
𝑧 − 5 = 9 and y = 7
y = 7
x + y = 8
x + 7 = 8 ⇒ x = 1
z – 5 = 9 ⇒ z = 14

23. a = 5
b = 7
c = 4
d = 9

24.  Addition of two matrices is possible only if both the matrices of
same order.
 sum of A + B = D where D = [dij] &
dij = aij +bij for all values of i and j.
Eg :
8 2
9 7
+
3 6
9 5
=
8 + 3 2 + 6
9 + 9 7 + 5
=
11 8
18 12

25.  A = [aᵢj] and k is any scalar.
 kA = k [aij]m × n = [k (aij)]m × n.
 Eg: A=
5 7
2 4
and k = 3
3A =
3𝑿5 3𝑿7
3𝑿2 3𝑿4
=
15 21
6 12

26.  –A is the Negative of the Matrix A
ie., –A = (– 1) A.
 Eg: A =
3 7 9
4 3 −2
- A = (-1)
3 7 9
4 3 −2
=
−3 −7 −9
−4 −3 2

27.  Difference of two matrices is possible only if both the
matrices of same order.
 A - B = D where D = [dij] &
dij = aij - bij for all values of i and j.
Eg :
8 2
9 7
-
3 6
9 5
=
8 − 3 2 − 6
9 − 9 7 − 5
=
5 −4
0 2

28.  (i) Commutative Law: A + B = B + A.
 (ii) Associative Law: (A + B) + C = A + (B + C).
 (iii) Existence of additive identity:
A + O = O + A = A,
where O is the zero matrix
 (iv) The existence of additive inverse:
A + (– A) = (– A) + A= O.
– A is the additive inverse of A.

29.  If A & B be two matrices of same order and k , l
are scalars, then
(i) k(A +B) = k A + kB
(ii) (k + l)A = k A + l A

30. http://www.youtube.com/watch?v=bo_5FTjc2Is
http://www.youtube.com/watch?v=0L90Kkn90J8

31.  The product of two matrices A and B is defined
only if the number of columns of A(pre
multiplier) is equal to the number of rows of B
(post multiplier).
 Let A = [aij]mxn and B = [bjk]n×p matrix.
Then AB = C = [cik](m×n)( nxp)= [cik]m×p
where cik = ai1 b1k + ai2 b2k + ai3 b3k + ... + ainbnk
= 𝑗=1
𝑛
𝑎𝑖𝑗 𝑏𝑗𝑘

32.  (1) Associative : (AB)C = A(BC), possible only
when equality defined on both sides.
 (2) Distributive: (i) A(B+C) = AB + BC
(ii) (A+B)C = AC + BC
 (3) Existence of Identity: For every square
matrix there exist an identity matrix.

33.  Matrix obtained by interchanging the rows and
columns is Transpose of a Matrix
 It is denoted by A’ or AT
 ie, A = (𝑎𝑖𝑗)mx n then AT = (aji)nxm

34.  (i) (A’)’ = A
 (ii) (kA)’ = kA’
 (iii) (A + B)’ = A’ + B’
 (iv) (AB)’ = B’A’

35.  A Square Matrix is Symmetric if A’ = A
 A Square Matrix is Skew Symmetric if A’ = -A
 A + A’ is Symmetric
A – A’ is Skew Symmetric
 Any Square Matrix can be expressed as the
sum of a symmetric and a skew symmetric
matrix

36.  Interchanging any two rows or two columns.
ie. Ri ↔ Rj (or) Ci ↔ Cj
 Multiplication of the elements of any row or column by a non-zero
number.
ie. Ri → k Rj (or) Ci → kCj
 Addition to the elements of any row or column, the corresponding
elements of any other row or column multiplied by any non zero number.
ie. Ri → Ri + kRj (or) Ci →Ci + kCj

37.  If A & B are Square Matrices such that
AB = BA = I
Then B is called inverse matrix of A
B = A-1
A is said to be invertible
 If A and B are invertible matrices of the same
order then (AB)-1 = B-1A-1