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7/28/2019 1.Introduction to Matrix Algebra
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I INTRODUCTION TO MATRIX A L GEB RA
Introduction to Matrix Algebra
INTRODUCTION TO MATRICESReference : Croft, A., & Davison, R. (2008). Mathematics for
Engineers - A Modern Interactive Approach, Pearson
Education.
A matrix is a rectangular array or block of numbers usually
enclosed in brackets.
A m x n matrix has m rows and n columns.Page 1
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Introduction to Matrix Algebra
If the matrix A has m rows and n columns we can write:
where aij represents the number or element in the ith row and
jth column.
=
mnmm
n
n
aaa
aaaaaa
A
21
22221
11211
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Introduction to Matrix Algebra
Special Matrices
A square matrixhas the same number of rows as columns.
The main diagonalof a square matrix is the diagonal
running from top left to bottom right.
An identity matrix, denoted by I , is a square matrix with
ones on the main diagonal and zeros elsewhere.
The transpose of A is obtained by writing rows as columns
and columns as rows, and is denoted AT.
Page 3
=100010
001
I
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Introduction to Matrix Algebra
Equality of Matrices
If A = (aij) and B = (bij), A = B if and only if aij= bij.
Addition and Subtraction of Matrices
Matrices of the same size may be added to and subtracted
from one another. To do this, the corresponding elements
are added or subtracted.
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Introduction to Matrix Algebra
e.g. 1 If
findA + B, B + C and B - C.
A + B is not defined asA andB are of not the same size.
B + C =
B C =
=
=
=
5
9
6
1
3
7
,
24
12
53
,203
412CBA
Page 5
=
+
75
85
1110
5
9
6
1
3
7
24
12
53
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Introduction to Matrix Algebra
Multiplication of a Matrix by a Number
Any matrix can be multiplied by a number. To do this, each
element of the matrix is multiplied by that number.
e.g.2 If , find 2A, -A.
2A =
-A =
=
8114
289
5137
A
Page 6
=
16228
41618102614
8*211*24*2
2*28*29*25*213*27*2
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Introduction to Matrix Algebra
Multiplication of Matrices
If A is a n x m matrix and B is a p x q matrix. For the product
AB to exist we must have m = p.
BAqpmn
Page 7
Note that matrix multiplication is :
i. not commutative (i.e. AB BA).
ii. associative [i.e. ABC = (AB)C = A(BC)].
iii. If C = AB, the element cij is found from row i of A andcolumn j of B, as follows:
c a bij ik kjk
n
==
1
ifm=p Cqn
=
ifm pdoes not exist
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=
=
2829
3626
3337
32
13
21
554
628
674
AB
Introduction to Matrix Algebra Page 8
262*63*21*8i.e.
3
1
1221=++==
=kkkbac
33 3 2 3 2
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Introduction to Matrix Algebra
Note that when a square matrix is post- or pre-multipliedby an identity matrix of the appropriate size the matrix is
unchanged, i.e.
AI = IA = A
Page 9
A B=
=
2 1 4
3 0 2
3 5
2 1
4 2
e.g. 3 If & , findAB.
AB=
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Introduction to Matrix Algebra
DETERMINANTS, INVERSE OF A MATRIX
Reference : Croft & Davison, Chapter 12, Blocks 3,4
Determinant
All square matrices, A, possess a determinant denoted by :
det(A), |A|.
Determinant of a 2 x 2 matrix
=
dc
baAIf , then det(A) = |A| = = ad - bc
dc
ba
A matrix which has a zero determinant is called singular.
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Introduction to Matrix Algebra
Minors and Cofactors of a 3 x 3 Matrix
Let aij be an element of a matrix A.
The minorof aijis the determinantformed by crossing out the ith
row and jth column of det(A).
The cofactorof aij= (-1)i+j x (minor of a
ij)
Note that the term (-1)i+j is called the place sign of the element
on the ith row and jth column. The following may help you to
memorize this.
++
+++
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Introduction to Matrix Algebra
Determinant of a 3 x 3 Matrix
Consider a general 3 x 3 matrix, A =
det(A) can be calculated by expanding along any row or
column. For example, expanding along the first row:
333231
232221
131211
aaa
aaa
aaa
|A| = a11*(its cofactor) + a12*(its cofactor) + a13*(its cofactor)
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Introduction to Matrix Algebra Page 13
e.g.1 Find the value of and
241
111
312
15316
52411
1741
145*33*1)2(*2
41
11*3
21
11*1
24
11*2
241
111
312
=++=
+
+
=
=
1531652411
1741
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Introduction to Matrix Algebra
Alternatively, by Rule of Sarrus
Repeat the 1st and 2nd column to right hand side of 3rdcolumn to form a 3 x 5 matrix.
det(A) =Add the product ofSOLID diagonals from left top to
right bottom and subtract the products ofDASH diagonalsfrom left bottom to right top.
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Introduction to Matrix Algebra Page 15
14
)2(*1*14*1*21*)1(*3
4*1*31*1*1)2(*)1(*2
4124111111
12312
241111
312
=
++=
=
Hence
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Introduction to Matrix Algebra
Properties of Determinants
i. If every element of a given row (or column) of the square
matrix is multiplied by the same factor, the value of thedeterminant is multiplied by that factor
ii. If |B| is obtained by interchanged any 2 rows (or columns) of
|A|, then |B| = -|A|.
iii. Adding or subtracting a multiple of one row (or column) to
another row (or column) leaves the determinant
unchanged.
iv. If A and B are 2 square matrices and that AB exists, then
det(AB) = det(A)det(B).
v. If 2 rows or 2 columns of a square matrix are equal, thedeterminant of the matrix is zero.
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Introduction to Matrix Algebra
Inverse of a Matrix
The inverse matrix of a square matrix A, usually denoted by A-1,
has the property :AA-1=A-1A = I
Note that if |A| = 0, A does not have an inverse.|A| 0, A has an inverse
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Introduction to Matrix Algebra
Finding the Inverse of a Matrix
The followings are steps to find the inverse of a matrix A when
|A| 0,
i. Find the transpose of A, denoted AT.
ii. Replace each element of AT by its cofactor. The resulting
matrix is called the adjoint of A, denoted adj(A).
iii.
A
AadjA
)(1 =
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Introduction to Matrix Algebra Page 19
e.g. 2 Find the inverse of
det(A) =14
=
241
111
312
A
=
=
=
375
173
4142
14
1
375
1734142
11
12
11
32
11
31
41
12
21
32
24
31
41
11
21
11
24
11
)(
1A
Aadj
T
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Introduction to Matrix Algebra Page 20
e.g. 3 Find the inverse of .
361
125
013
=B
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