It contains the basics of matrix which includes matrix definition,types of matrices,operations on matrices,transpose of matrix,symmetric and skew symmetric matrix,invertible matrix,
application of matrix.
2. INDEX
1.MATRIX
I . Order of matrix
2.TYPES OF MATRIX
I . Column matrix
II . Row matrix
III . Square matrix
IV . Diagonal matrix
V. Identity matrix
VI . Null matrix
3.OPERATIONS ON MATRIX
I . Addition and subtraction
II . Multiplication
4.TRANSPOSE OF A MATRIX
5.SYMMETRIC AND SKEW
SYMMETRIC MATRIX
6.INVERTIBLE MATRIX
7.APPLICATION OF MATRIX
3. MATRICES
A matrix is a structural representation of rows and
columns which is enclosed within two brackets.
dc
ba
03
24
11
The horizontal lines of elements are said to
constitute, rows of the matrix and the vertical lines of
elements are said to constitute, columns of the matrix.
2x2 2x2
1x2
5. TYPES OF MATRICES
1. Column matrix or vector:
2
4
1
3
1
1
21
11
ma
a
a
A matrix is said to be a column matrix if it has
only one column. In general, A = [aij] m × 1 is a
column matrix of order m × 1.
2x13x1
6. 611 2530
naaaa 1131211
2. Row matrix or vector:
A matrix is said to be a column matrix if it has
only one Row. In general, A = [aij] 1 × n is a
column matrix of order 1 x n.
1x3 1x4
7. A matrix, in which the number of rows is equal to the
number of columns, is said to be a square matrix. Thus
an m × n matrix is said to be a square matrix if m = n
and is known as a square matrix of order ‘n’.
3. Square matrix
03
11
166
099
111
2x2 3x3
8. 4. Diagonal matrix
A square matrix B = [bij] m × m is said to be a
diagonal matrix if all its non diagonal elements are
zero. That is a matrix B = [bij] m × m is said to be a
diagonal matrix if bij = 0, when i ≠ j.
100
020
001
9000
0500
0030
0003
3x3 4x4
9. 5. Unit or Identity matrix
A square matrix in which elements in the
diagonal are all 1 and rest is all zero is
called an identity matrix.
1000
0100
0010
0001
10
01
2x2
4x4
10. 6. Null (zero) matrix
A matrix is said to be zero matrix or null
matrix if all its elements are zero. We
denote zero matrix by O.
0
0
0
000
000
000
3x1 3x3
11. EQUALITY OF MATRICES
Two matrices are said to be equal only when all
corresponding elements are equal
Therefore their size or dimensions are equal as well.
A =
325
012
001 B =
325
012
001 A = B
3x3 3x3
12. OPERATIONS ON
MATRICES
ADDITION AND SUBTRACTION OF MATRICES
The sum or difference of two matrices, A and B of the
same size yields a matrix C of the same size.
ijijij bac
Matrices of different sizes cannot be added or
subtracted
14. PROPERTIES OF MATRIX ADDITION
Commutative Law:
A + B = B + A
Associative Law:
A + (B + C) = (A + B) + C = A + B + C
Existence of additive identity
A + 0 = 0 + A = A
The existence of additive inverse
A + (-A) = 0
15. SCALAR MULTIPLICATION OF MATRICES
Matrices can be multiplied by a scalar (constant or
single element)
Let k be a scalar quantity; then
kA = Ak
416
128
48
412
4
14
32
12
13
14
32
12
13
4
4x24x2 4x2
16. MULTIPLICATION OF MATRICES
The product of two matrices A and B is defined if the
number of columns of A is equal to the number of
rows of B.
)37()22()84()57()62()44(
)33()22()81()53()62()41(
35
26
84
724
321
5763
2131
2x2
3x2
2x3
2x2
18. TRANSPOSE OF A
MATRIX
If A = [aij] be an m × n matrix, then the matrix
obtained by interchanging the rows and columns of
A is called the transpose of A. Transpose of the
matrix A is denoted by A′.
135
742
A
Then transpose of A, denoted AT is:
17
34
52
T
A
2x3
3x2
19. SYMMETRIC
MATRICES
A Square matrix is symmetric if it is equal to
its transpose:
A = AT
741
45.13
132
A
741
45.13
132
A
3x3
3x3
20. SKEW SYMMETRIC
MATRICES
A square matrix A is said to be a skew symmetric if
B’ = – B . And all elements in the principal diagonal
of a skew symmetric matrix are zeroes.
0
0
0
gf
ge
fe
B
3x3
21. IMPORTANT POINT
i. For every square matrix A, A+A' is a symmetric matrix and
A+A' is a skew symmetric matrix.
ii. Any square matrix can be expressed as the sum of a
symmetric and a skew symmetric matrix
Let A be a square matrix, then we can write
A=1/2(A+A')+1/2(A-A')
22. INVERTIBLE
MATRICES
For example:-
A square matrix of order n is invertible if there exists
a square matrix B of the same order such that AB =
I = BA, Where I is identify matrix of order n.
12
25
B
52
21
A
2x2
2x2
