Matrices
A matrix is a rectangular array of numbers.
Dimension = no. of rows, and no. of columns,
i.e., m x n - read as m by n
10
12
15
20
20
15
2x3
Matrices
Position of an element
a11
a21
a31
a12
a22
a32
a13
a23
a33
Matrices
Row Vector
Column Vector
Square matrix
Identity Matrix or Unit matrix – a square matrix having all
the elements 0 except primary diagonal values are 1.
Matrix Operations
Scalar Multiplication
a
b
c
5
5a
5b
5c
5c
5b
5d
=
c
b
d
Matrix Operations
Addition and Subtraction ( dimension should be same)
a
b
c
5a
5b
5c
5c
5b
5d
+
c
b
d
6a
6b
6c
6c
6b
6d
=
Matrix Operations
Commutative Law: A + B = B + A
Associative Law: A + (B + C) = (A + B) + C
Zero Matrix
0
0
0
0
0
0
A – 0 = A and 0 – A = - A
A + 0 = A (Identity law)
Matrix Operations
Matrix multiplication
Let A is a matrix of dimension m x n, and B is a matrix
of dimension p x q; then for matrix multiplication A .
B, n must be equal to p.
5
A = (3
2) and B =
A . B = 23
4
Matrix Operations
1
3
21
47
2
4
2x2
.
24
54
5
8
6
9
27
61
7
10
2x3
Matrix Operations
If A . B is possible, B . A is not possible except for the
case when both are square matrix of same
dimension.
But,
A.B
B.A
Row Operations
1. Can be multiplied or divided a row by a non-zero
constant
2. Can be added a multiple of one row to a multiple of
another row
3. Two rows can be interchanged
Inverse Matrix
A . A-1 = I = A-1 . A
Two methods for finding inverse matrix:
1. Gauss and Jordan elimination method
2. Formula method : A-1 = Adjoint of A/Determinant of A
A=
3 2
1 1
A-1 =
1
-2
-1
3
Transpose of a Matrix
3
2
1
1
3
1
2
1
A=
AT =
Determinant of a Square Matrix
Determinant of order or dimension 2
a
b
c
d
= ad - bc
Determinant of order or dimension 3
a b
c
d e
f
g h
i = a (e.i – h.f) – b (d.i – g.f) + c (d.h – g.e)
Adjoint of a Square Matrix
Find the adjoint of the matrix
1
1
1
1
2
-3
2 -1
3
c11
Adj A = Transpose of
c21
c31
c12
c22
c32
c13
c23
c33
Matrices
c11 = cofactor of a11 = + (2 x 3 - (-1) x (-3)) = 3
c12 = cofactor of a12 = - (1 x 3 - 2 x (-3)) = - 9
c13 = cofactor of a13 = + (1 x (-1) - 2 x 2) = - 5
c21 = cofactor of a21 = - (1 x 3 - (-1) x 1) = - 4
Matrices
Adj A = Transpose of
=
3
-4
-5
-9
1
4
-5
3
1
3
-9
-5
-4
1
3
-5
4
1
Matrices
Determinant of A =
1 2 -3
-1 1
-1 3
2
-3
3
+1 1
2
2
-1
= 1 (2x3 – (-1)x(-3)) – 1 (1x3 – 2x(-3)) + 1 (1x (-1) – 2x2))
= -11
Matrices
A-1 = Adjoint A / Determinant of A
3
-9
-5
-4
1
3
=
-11
-5
4
1
Matrices
2x1 + 3x2 = 17
x1 + 2x2 = 10
2x1 + 2x2 + 3x3 = 3
x2 + x3 = 2
x1 + x2 + x3 = 4
Matrices
2
3
A=
x1
X=
1
2
17
B=
x2
10
Matrices
2 2 3
0 1 1
1 1 1
x1
x2
x3
=
3
2
4
Matrices
Two methods for solving the equations:
1. X = A-1 B
Because, A -1 A X = A-1 B
2. x1 = Nx1/ D, x2 = Nx2/D, and x3 = Nx3/D
where D is the determinant of the matrix, Nx is the
determinant of the matrix after replacing the respective
column by the B matrix.
Matrices
2x1 - x2 = 5
3x1 + 2x2 = -3
(D = 7, Nx1= 7, Nx2 = -21)
2x1 + x2 - x3 = 3
x1 + x2 + x3 = 1
x1 - 2x2 - 3x3 = 4 (D = 5, Nx1= 10, Nx2 = -5, and Nx3 = 0)