-
Topic 7
Matrices
-
Definition A matrix is a rectangular array of numbers.
A matrix with m rows and n columns is called an m x n matrix. The matrix is said to have order m x n. The entries in a matrix are called the elements of the matrix.
-
Definition
nmAn matrix A has the form
mnmmm
n
n
n
aaaa
aaaaaaaaaaaa
.....................................
......
....
....
321
3333231
2232221
1131211
A =
The elements are enclosed in large square brackets.
-
Example
Identify orders of the following matrices:
0 32 11 4
[ ]2, 3, 5, 13 5 0 42 3 1 3
1 2 1 12 4 3 26 0 1 34 2 3 5
3 2 1 4 2 4 4 4
-
Example
Write down the 3 3 matrix with elements given by .ij ija a i j =
11 12 13
21 22 23
31 32 33
a a aa a aa a a
-
Special Matrices --- Row Matrix
A matrix with only one row is called a row matrix.
[ ]1 2 3 4
-
Special Matrices --- Column Matrix
A matrix with only one column is called a column matrix.
1234
-
Definition
mnmmm
n
n
n
aaaa
aaaaaaaaaaaa
.....................................
......
....
....
321
3333231
2232221
1131211
A =
1, 2: ( ,..., )i i i inz a a a=
1z2z3z
mz
1
2
m
zz
A
z
=
-
Definition
mnmmm
n
n
n
aaaa
aaaaaaaaaaaa
.....................................
......
....
....
321
3333231
2232221
1131211
A =
1
2:
j
jj
mj
aa
s
a
=
1s 2s 3s ns
1 2( , ,..., )nA s s s=
-
Special Matrices --- Square Matrix
A matrix where the number of rows is equal to the number of columns is called a square matrix.
1 2 1 12 4 3 26 0 1 34 2 3 5
-
Special Matrices --- Symmetric Matrix
1 2 62 4 36 3 1
A square matrix such that for all values of and is called a symmetric matrix.ij jia a i j=
-
Example
Find the value of such that the following matrix is symmetric.0 4 5
2 5 75 7 6
xx
x+
2 4x x= +
4x =
-
Special Matrices --- Diagonal Matrix
A square matrix, in which all the elements not on the principal diagonal are zeros, is called a diagonal matrix.
1 0 0 00 4 0 00 0 1 00 0 0 5
-
Special Matrices --- Identity Matrix
A diagonal matrix, in which every diagonal element is 1, is called an identity matrix.
An nxn identity matrix is denoted by In or simply I.
1 0 0 00 1 0 00 0 1 00 0 0 1
The role of identity matrices in matrix world is just like
the role of 1 in real numbers.
-
Special Matrices --- Zero Matrix
A matrix where every element is 0 is called a zero matrix. An mxn zero matrix is denoted by 0mn or 0 if the order is obvious
from the context. The zero matrix is also called the null matrix.
0 0 0 00 0 0 00 0 0 0
Note: zero matrix is not necessary a square matrix.
-
Special Matrices --- Transpose of Matrix
The transpose AT of an mXn matrix whose rows are the corresponding columns of A.
1 02 14 7
A
1 2 40 1 7
-
Theorem
( )TTA A=
is a symmetric matrix.A TA A=
-
Matrix Operations
-
Operations Btw Matrices We can ADD, SUBTRACT and
MULTIPLY matrices. However, unlike ordinary numbers, these
operations are dependent on the order of the matrices involved.
There is no DIVISION of matrices.
-
Definition --- Equality Two matrices equal to each other if they have
the same order and all elements in the same position of the matrices equal to each other.
=
21
yx 1=x 2=yIf , then and
-
Operation 1 - Addition
=
+++
+++
+++
=
+
1531276121083
783066075175914412
736057941
806715142
=
++
++=
+
141185
410382614
4321
10864
2 4 1 15 1 7 3 ?6 0 8 4
+ =
-
Operation 2 - Subtraction
=
=
+
130742801
783066075175914412
736057941
806715142
=
=
6543
410382614
4321
10864
-
Operation 3 Scalar Multiplication
IMPORTANT NOTE
When a matrix is multiplied by a scalar (i.e. a number, whether integer or non-integer), EVERY element in the matrix is multiplied by that scalar
Example:-
=
=
8642
42322212
4321
2
-
Definition
1
10
: ,0
0
e
=
2
01
: ,0
0
e
=
00
: 0
1
ne
=
-
Operation 4 Matrix Multiplication
IMPORTANT NOTE
When a matrix A is multiplied by another matrix B, it is first important to ensure that the no. of columns in matrix A is EQUAL to the number of rows in matrix B. Otherwise, A and B cannot be multiplied.
-
Example
[ ]1
2 5 7 36
[ ]2 1 5 3 7 6= + +
[ ] [ ]2 15 42 59= + + =Order 1X3
Order 3X1
(1X3) X (3X1) = 1X1
Order 1X1
-
Example
[ ]
=
=
541108215123
514111524212534313
541123
3X1 1X3 3X3
-
Exercise
2 1 31 2
(1) 1 0 1 ?0 3
4 3 1
=
1 2 1 2 3(2) ?
0 3 -1 3 5
=
-
Matrix operations Matrix Multiplication
IMPORTANT properties of matrix multiplication
Theorem 3 Let A, B, C, 0(zero matrix) and I(identity matrix) be conformable matrices and k be any scalar. 1. (AB)C = A(BC) 2. A(B + C) = AB + AC 3. k(AB) = (kA)B = A(kB) 4. A0 = 0 5. AI = A 6. (AB)T = BTAT
-
Matrix operations Matrix Multiplication
IMPORTANT NOTE
(1) Matrix multiplication is NOT commutative, i.e.: AB BA
Example:-
=
161143
4321
3210
3210
4321
Let A = and B = Then AB =
=
15874
3210
4321
but BA =
Hence AB BA
-
Matrix operations Matrix Multiplication
IMPORTANT NOTE
(2) Cancellation law DOES NOT apply, i.e.: If AB = AC, it DOES NOT imply B = C
Example:-
1111Let A =
1111, B =
0220and C =
=
2222
1111
1111
Then AB =
=
2222
0220
1111
and AC =
So AB = AC but B C
-
Practice
2
1 2 3Given 2 3 4 , find 3 4 5 .
3 4 5A A A I
= +
-
Linear System of Equations (LSE)
11 1 12 2 1 1
21 1 22 2 2 2
1 1 2 2
......
...
n n
n n
m m mn n m
a x a x a x ba x a x a x b
a x a x a x b
+ + + =
+ + + =
+ + + ={
11 12 1 1 1
21 22 2 2 2
1 2
n
n
m m mn n m
a a a x ba a a x b
a a a x b
=
Ax b=
-
Determinants
-
Determinants (Introduction)
Determinant is a special measurement/property of square matrix.A square matrix is a matrix with the same number of rows and columns.
Determinant is denoted by: a b cd e fg h i
=
If a determinant has rows and columns, we say it is an th order determinant.n n n
-
Evaluating 2nd order determinants
Start with the TOP LEFT corner and multiply with BOTTOM RIGHT corner (1)
Take TOP RIGHT corner and multiply with BOTTOM LEFT corner (2)
Take (1) (2).
a bad bc
c d=
-
Example
2 3Evaluate:
1 4
2 3 2(4) 3( 1)
1 4=
8 3= +
11=
-
Minor of an element
The minor of an element aij is the determinant formed after removing row i and column j containing the element,
And then forming a determinant with the FOUR remaining elements
For example, to find the minor for a12:-
11 12 13
21 22 23
31 32 33
a a aa a aa a a
= minor of a12 21 23
21 33 23 3131 33
a aa a a a
a a= =
-
Example 1
Given that the minor of 5 in the determinant is 12, determine the value of x.
68527312
x
41216
12)1)(()8(2812
5 ofMinor
=
=+
===
xx
xx
-
-
Cofactor of an element
The notation Aij is used to denote the cofactor of an element aij .
The relationship between the minor and cofactor is
The only difference between the minor and cofactor is that the cofactor has the (-1)i+j term.
Thus, to find the cofactor, it is necessary to FIND THE MINOR FIRST.
cofactor of aij = (1)i+j(minor of aij )
-
Example 2
Find the cofactors for 8 and 3 respectively in the determinant given.
684527312
( ) [ ]
[ ] 112110
)7)(3()5(25732
18 ofCofactor 23
==
== +
( ) [ ]
[ ] 48856
)4)(2()8(78427
13 ofCofactor 31
==
=
= +
-
Evaluating 3rd Order Determinant
11 12 13
21 22 23
31 32 33
a a aa a aa a a
11 11 12 12 13 13a A a A a A= + +
21 21 22 22 23 23a A a A a A= + +
31 31 32 32 33 33a A a A a A= + +
11 11 21 21 31 31a A a A a A= + +
12 12 22 22 32 32a A a A a A= + +
13 13 23 23 33 33a A a A a A= + +
A determinant can be evaluated via any row or any
column
This process is called Laplace
Expansion
-
Example 3
Expand along row 3, using Laplace expansion
3 1 3 2 3 36 8 4 8 4 61( 1) ( 3)( 1) 2( 1)5 1 2 1 2 5
+ + + = + +
4 6 82 5 11 3 2
4 6 82 5 11 3 2
86163634
)8(2)12(334]1220[2]164[3]406[
=
++=
++=
++++=
-
Example 4
2 0 3Evaluate 123 1 876 using Laplace expansion.
2 0 5
2 2 2 31 ( 1)2 5
+=
2(5) 2(3)=
4=
-
Sarrus method
Note that Sarrus method applies only for 3rd order determinants (i.e. determinants of 3x3 matrices)
a b cd e fg h i
=
a b cd e fg h i
aei bfg+ dhc+ ceg bdi ahf
-
Example 5
3 0 4Solve for given that 1 1 2 19.
6 5x x
x+ =
Using Sarrus' method to expand the determinant, we have
3( 1)( 5) 0( 2)(6) 1( )(4) 4( 1)6 3( 2) 0(6)( 5) 19x x x x+ + + + =
15 15 4 24 24 6 19x x x x + + =
29 39 19x =
29 58x =
2x =
-
Evaluating higher order determinants
-
Laplace expansion
= a21A21+a22A22+a23A23++a2nA2n
= a11A11+a21A21+a31A31++an1An1
And so on
11 12 13 1
21 22 23 2
11 11 12 12 13 13 1 131 32 33 3
1 2 3
...
n
n
n nn
n n n nn
a a a aa a a a
a A a A a A a Aa a a a
a a a a
= + + + +
LLLM M M O MLNote that the expansion can be done along ANY row or column the answer should be the same regardless
-
Inverse of a matrix
Given a matrix A, we can find A-1 such that AA-1 = I, where I is the identity matrix.
BUT, this is only possible if
(1) A is a square matrix
(2) the determinant of A is non-zero
Theorem The inverse A-1 of an matrix A is given by
A-1 adj A where is the determinant of A.
A1
=
A
-
Inverse of a matrix
The adjoint of a matrix A, denoted by adjA is the transpose of the cofactors of A, and is found by:-
333231
232221
131211
aaaaaaaaa
If A =
T
AAAAAAAAA
333231
232221
131211then adj A =
332313
322212
312111
AAAAAAAAA
=
.
where is the cofactor of the element ijA ija
-
Example
Given a 3X3 matrix A which satisfies equation
Find A2 3A+ I = 0
A1
-
Linear System of Equations (LSE)
11 1 12 2 1 1
21 1 22 2 2 2
1 1 2 2
......
...
n n
n n
m m mn n m
a x a x a x ba x a x a x b
a x a x a x b
+ + + =
+ + + =
+ + + ={
11 12 1 1 1
21 22 2 2 2
1 2
n
n
m m mn n m
a a a x ba a a x b
a a a x b
=
Ax b=
