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MATRICES
Definition
A matrix is an array of numbers or symbols in rows or columns. A matrix is written
between brackets, e.g.
2 3 1
0 4 7
is a 2 x 3 matrix since it has 2 rows and 3 columns. In general m x n matrix has m
rows and n columns.
A 1 x n matrix is called a row vector
(a b c)
a
b
c
A m x 1 matrix is called a column vector
If m = n the matrix is square
If m  n the matrix is rectangular
We represent the general matrix in the following form:
 a11 a12 a13 
 a21 a22 a23 
 a31 a32 a33 
The first subscript determines the row and the second the column in which the
element is placed. A matrix in which all the elements are zero is called a NULL
matrix. For example:
 0 0
 0 0
0 0
is a 3 x 2 null matrix.
1
Equality of matrices
Two matrices are equal if and only if they are the same order and corresponding
elements are equal. For example if
 a b c  =  3 0 -1
d e f 2 7 5 
2x3
2x3
a = 3, b = 0, c = -1, d = 2, e = 7, f = 5
Addition of matrices
If A and B are m x n matrices the result of adding (or subtracting) A + B is an m x n
matrix in which the elements are the sum (or difference) of the corresponding
elements of A + B.
For example
 3 5  -  6 2  =  -3 3 
 1 4   1 8   0 -4 
Two matrices A and B are said to be conformable for addition and subtraction only if
they are of the same order.
Matrix Multiplication
Two matrices A and B are said to be conformable for multiplication in the form AB
only if the number of columns of A is equal to the number of rows of B.
Also AB = C
If A is an m x n matrix and B is n x r then C = m x r matrix.
The multiplication procedure is :
 a11 a12 a13   b11 b12 
 a21 a22 a23  x  b21 b22 
2x3
 b31 b32 
3x2
 [a11b11
+ a12b21 + a13b31]
 [a21b11 + a22b21 + a23b31]
+ a12b22 + a13b32] 
[a21b12 + a22b22 + a23b32] 
[a11b12
2x2
2
If A = 
2 0
 1 4
and B = 
3 4
 -2 1 
find AB and BA
AB
 2 0   3 4
 1 4 -2 1 
=
(2 x 3 + 0 x-2) (2 x 4 + 0 x 1)
(1 x 3 + 4 x –2) (1 x 4 + 4 x 1)
=  6 8
-5 8
Find BA
 3 4 2 0  =
-2 1  1 4
Note AB  BA
Exercise
1.
If A = 
1 3 2
 0 0 4
&B
=  1 2 1 0
0 1 4 1
 -1 2 1 3 
3
2.
If A = 
1 2 1
3 4 6
&B
=0 1
1 1
1 1
Find AB
3. If A =
4.
1 1 2
 0 1 1
1 0 3 
and B = 
a
b
c
Find the simultaneous linear equations which are given in a matrix form by
 1 1 2  x  =  3 
 0 1 1  y   5 
 1 0 3   z  -2 
4
Answers to Exercises 1 to 4
1.
2.
3.
4.
3 4
 10 14
5
2
7
 3x -2y + 2z  =  3 
 2x + y - z 
5
 x – 5y – 3z  -2 
 6 9 12 
 16 23 30
 26 37 48 
5
Special Matrices
Diagonal Matrices
A diagonal matrix is a square matrix in which all the elements are zero except those in
the principal diagonal. For example:
1
0
 0
0
0 0
3 0
0 –7
0 0
0  is a 4 x 4 diagonal matrix.
0
0
4
Unit matrix denoted by I, is a diagonal matrix in which all the elements of the
principal diagonal are all unity. For example
I=
 1 0 0  is a 3 x 3 matrix
 0 1 0
0 0 1
If A is any square matrix and I is the unit matrix of the same order .
AI = IA = A
For example
 3 -1   1 0  =  (3 x 1 + -1 x 0) (3 x 0 + -1 x 1) 
2 4 0 1
 (2 x 1 + 4 x 0)
( 2 x 0 + 4 x 1) 
=  3 -1
2 4 
 1 0  3 -1 =  3 -1
 0 1  2 4   2 4 
Exercise
Find A if A = (B + C)D and
B=
2 -1 3
 -1 2 -3
C=
-1 3 -1 D =  0 -2 4
 -2 1 0
 1 3 -4
 -1 1 2
6
Transpose of a matrix
The transpose of a matrix is the matrix obtained by interchanging rows and columns
in order. The transpose of A is denoted by A
For example
A=
 3 1
0 4
 5 -7 
and A = 
3 0 5
 1 4 -7
Determinants
A determinant is an array of numbers or symbols in rows and columns and is written
between vertical lines.
a
c
1.
2.
3.
4.
5.
6.
7.
b
d
or
1 3 5 or a11 a12 a13
2 1 4
a21 a22 a23
0 3 7
a31 a32 a33
The numbers or symbols are called elements
Any horizontal set of elements is called a row
Any vertical set of elements is called a column
A determinant is always square i.e. no of rows = number of columns
For subscripted elements aij i denotes the row and j the column
The elements shown arrowed form the principle diagonal
The first element in the first row (and column) is called the leasing element.
The order of Determinants
The order of a determinant is equal to the number of rows or column in the
determininant. The above deternmininats are 2nd , 3rd and 3rd order respectively.
Minors and Co-factors
Minors
Every element in a determninant has a minor. The monor of an element
is the determinant obtained by striking out the row and column containing that
element. For example consider:
a b
d e
g h
c
f
i
The minor of d is
The minor of I is
b
h
a
d
c
i
b
e
7
The minor of h is
a
d
c
f
Co-Factors
The co-factor of an element is called the minor with the correct sign attached to it.
The sign depends upon the position of the element. The sign convention is:
+ - +
- + + - +
For example consider
a b c
d e f
g h i
The co-factor of d is –
b c
h i
d is negative according to the sign position
The co-factor of g is +
b c
e f
g is positive according to the sign position
The co-factor of f is –
a b
g h
f is negative according to the sign position
Expansion and Evaluation of a determinant
A determinant can be expanded about any row or column.For example row 1
a1 b1 c1 = a1 b2 c2 - b1 a2 c2 +c1 a1 b2
a2 b2 c2
b3 c3
a3 c3
b3 c3
a3 b3 c3
column 2 expanded gives:
- b1 a2 c2 + b2 a1 c1 - b3 a1 c1
a3 c3
a3 c3
a2 c2
Also
b2 c2 = b2 x c3 – c2 x b3
b3 c3
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Examples
1
Evaluate
2
Evaluate
3.
=
3
1
2 = 3 x 4 – 2 x 1= 10
4
4 1 = 4 x 1 – 1 x –2 = 6
-2 1
1 0 2
3 4 -1
0 3 5
Evaluate 
About row 1
=
1 4 -1 - 0 3 -1 + 2 3
3 5
0 5
0
4
3
 = 1(4 x 5 –(-1 x 3) – 0 + 2(3 x 3 – 4 x 0 ) = 23 + 18 = 41
Exercises
1.
Evaluate
3
7
2 =
1
2
Evaluate
-3 -2 =
-1 4
3.
=
1 2 1
0 -3 4
3 5 1
Evaluate 
Answers
1.
2.
3.
-11
–14
10
9
Determinant of a square Matrix
The determinant of a square matrix is the determinant with the same elements as the
matrix. The determinant of A is denoted by det A or A.
 3 2
 4 -1
If A =
det A =
3 2 = (3 x –1) – (2 x 4) = -11
4 -1
 2 1 show that A -I = 0
 1 1
If A =
becomes 2 - 3 + 1 = 0
show that A satisfies the matrix equation
A2 – 3A + I = 0
Also if B is a matrix such that BA = I use this equation to find B
A -I =
 2 1 -  1 0 Note remember I is the unit matrix
 1 1
 0 1
=  2 1 -   0
 1 1  0 
=  2 -  1- 0
 1 - 0 1 - 
= 2-
1
1
1 - 
So A
-I = 2 - 
1
1
1-
= ((2 - ) x (1 - ) – 1 x 1)
= 2 - 2 -  + 2 – 1
10
= 1 - 3 + 2
So if A
Then 1
-I = 0
- 3 + 2 = 0
A2 =  2 1 2 1 =  5
(2 x 2 + 1 x 1)
 1 1 1 1  (1x 2 + 1 x 1) (1 x 1 + 1 x 1) 
=  5 3
 3 2
3A = 3 2 1 =  6 3
 1 1  3 3
So A2 – 3A + 1 =  5 3 -  6 3 + 1 0 
 3 2  3 3 0 1
=  0 0
 0 0
Now A2 – 3A + I = 0
And BA = I which is the unit matrix. Multiplying the above equation through by B
gives:
BA2 – 3BA + IB = 0 now BI = B remember multiplying by the unit matrix leaves the
matrix unchanged.
i.e. BAA – 3BA + B = 0 since BA = I the equation becomes
IA – 3I + B = 0
So A – 3I + B = 0
B = 3I – A
=3
1 0 - 2 1 = 3 0 - 2 1 = 1 -1
0 1 1 1 0 3 1 1 -1 2
11
Adjoint Matrix
The adjoint matrix of a square matrix A is the matrix whose elements are the cofactors of the corresponding elements of det A
Let A =  a1
b1 c1 
 a2 b2 c2 
 a3 b3 c3 
Now A =  a1 a2
a3 
 b1 b2 b 3 
 c1 c2 c3 
Adj A = +
b2 b3
c2 c3
- b1 b3 + b1 b2
c1 c3 c1 c2
- a2 a3 + a1 a3 - a1 a2
c2 c3
c1 c3
c1 c2
+ a2 a3 - a1 a3 + a1 a2
b2 b3
b1 b3
b1 b2
Examples
If A = 3
2 find adj A evaluate det A and A adj A
1 4
1.
A = 3
1 adj A = 4 -2
2 4
-1 3
det A
3 -2 = 12 – 2 = 10
1 4
A adjA = 3 2 4 -2 = (3 x 4 + 1 x-2) (3 x –2 + 2 x 3)
1 4 -1 3 (1 x 4 + 4 x-1) (1 x –2 + 4 x 3)
12
A adjA = 10 0 = 101 0
0 10
0 1
 AadjA = detA I
2.
Let A = 
1 2 0
 3 -1 4 
2 0 6
Now A = 
1 3 2
 2 -1 0 
 0 4 6
Adj A = +
-1 0
4 6
- 2
0
0 + 2 -1
6 0 4
- 3 2 +1
4 6
0
2 - 1
6
0
+ 3 2 - 1
-1 0
2
2 + 1 3
0
2 -1
So adj A =
-6 -12 8
-10 6 -4
2 4 -7
det A =
1 2
3 -1
2 0
det A = 1
3
4
0
4
6
-1 4 - 2 3 4 + 0
0 6
2 6
= 1(((-1 x 6) – 0 ) –2((3 x 6) – (4 x 2)))
= -26
13
AadjA =  1 2 0 
 3 -1 4 
2 0 6
-6 -12 8
-10 6 -4
2
4 -7
=
(-6 –20 + 0) (-12 –12 + 0) (8 –8 + 0 )
(-18 +10 + 8) (-36 -6 + 16) (24 + 4 - 28)
(-12 +0 - 12) (-24 +0 + 24) (16 + 0 - 42)
=
-26 0 0
0 -26 0
0 0 -26
= -26 1 0
0 1
0 0
0
0
1
= detAI
Exercises
1.
If A =
2.
A=
2 3
-1 1
2 -1
4 -4
0 2
find AdjA detA and evaluate AadjA
3
6
3
Find adjA
Answers
1.
5
0
0
5
2.
adjA =
-24 9 6
-12 6 0
8 -4 -4
14
Inverse of a Square matrix
If A and B are square matrices such that AB = I = BA
Then B is called the inverse of A and is written A-1. Now we have seen that
AadjA = det AI
So
1
= A adjA = I
detA
Hence A-1 = 1 x adjA
detA
Example
3
1
1. If A =
A =
5
3
3
5
1
3
3 -5
-1 3
adj A =
3
5
= 3 x 3 – (-5 x -1) = 4
detA =
1
3
A-1 = 1 x adjA =
det A
=
2.
find A-1
If A =
1
3
2
1 3 -5
4 -1 3
3/4 -5/4
-1/4 3/4
2 0
-1 4
0 6
15
6
-10
2
Adj A =
detA =
-12 8
6 -4
4 -7
-26
A-1 = -1
6
-12 8
26 -10
6 -4
2
4 -7
Exercise
If A =
1
2
3
2
3
3
Ans A-1 = 1
3
2
4
6
1
7 –2 -5
-3 3
find A-1
-5
4
1
16
Example Find the current in each branch of the network shown.
2
I1
4
I2
4V
7
4V
10V
Using Maxwell’s cyclic currents
-4 - 2I1 – 4(I1- I2) + 10 = 0
-4 - 6 I1 + 4I1 + 10 = 0
6 - 6 I1 + 4I1 = 0
6 = 6 I1 - 4I1----------------------------------------------------------(1)
-4 - 7I2 – 4(I2- I1)-10= 0
-4 + 4I1 - 11I2 - 10 = 0
-14 = -4I1 + 11I2-----------------------------------------------------(2)
In Matrix form
R IC = V
6 -4 I1 =  6
-4 11 I2 -14
R
IC
V
RIc = V
R-1RIc = R-1V
Ic = R-1V
17