PPT: Basics of Matrix & Operations on Matrix | Engineering Mathematics - Civil Engineering (CE) PDF Download

 Page 1


4.4 Matrices: Basic Operations 
Page 2


4.4 Matrices: Basic Operations 
Addition and Subtraction of matrices 
? To add or subtract matrices, they must be of the same order, 
mxn. To add matrices of the same order, add their 
corresponding entries.  To subtract matrices of the same order, 
subtract their corresponding entries. The general rule is as 
follows using mathematical notation: 
ij ij
ij ij
A B a b
A B a b
??
? ? ?
??
??
? ? ?
??
Page 3


4.4 Matrices: Basic Operations 
Addition and Subtraction of matrices 
? To add or subtract matrices, they must be of the same order, 
mxn. To add matrices of the same order, add their 
corresponding entries.  To subtract matrices of the same order, 
subtract their corresponding entries. The general rule is as 
follows using mathematical notation: 
ij ij
ij ij
A B a b
A B a b
??
? ? ?
??
??
? ? ?
??
An example: 
? 1. Add the matrices 
? First, note that each matrix 
has dimensions of 3X3, so 
we are able to perform the 
addition. The result is shown 
at right:
? Solution: Adding 
corresponding entries we 
have
4 3 1 1 2 3
0 5 2 6 7 9
5 6 0 0 4 8
??
? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
??
? ? ? ?
3 1 4
6 2 7
5 10 8
?
??
??
?
??
??
?
??
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4.4 Matrices: Basic Operations 
Addition and Subtraction of matrices 
? To add or subtract matrices, they must be of the same order, 
mxn. To add matrices of the same order, add their 
corresponding entries.  To subtract matrices of the same order, 
subtract their corresponding entries. The general rule is as 
follows using mathematical notation: 
ij ij
ij ij
A B a b
A B a b
??
? ? ?
??
??
? ? ?
??
An example: 
? 1. Add the matrices 
? First, note that each matrix 
has dimensions of 3X3, so 
we are able to perform the 
addition. The result is shown 
at right:
? Solution: Adding 
corresponding entries we 
have
4 3 1 1 2 3
0 5 2 6 7 9
5 6 0 0 4 8
??
? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
??
? ? ? ?
3 1 4
6 2 7
5 10 8
?
??
??
?
??
??
?
??
Subtraction of matrices
? Now, we will subtract 
the same two matrices
? Subtract corresponding 
entries as follows:
4 3 1 1 2 3
0 5 2 6 7 9
5 6 0 0 4 8
??
? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
??
? ? ? ?
4 ( 1) 3 2 1 3
0 6 5 ( 7) 2 9
5 0 6 ( 4) 0 8
? ? ? ? ?
??
??
? ? ? ? ?
??
?? ? ? ? ? ?
??
5 5 2
6 12 11
5 2 8
??
??
??
??
??
??
??
??
= 
Page 5


4.4 Matrices: Basic Operations 
Addition and Subtraction of matrices 
? To add or subtract matrices, they must be of the same order, 
mxn. To add matrices of the same order, add their 
corresponding entries.  To subtract matrices of the same order, 
subtract their corresponding entries. The general rule is as 
follows using mathematical notation: 
ij ij
ij ij
A B a b
A B a b
??
? ? ?
??
??
? ? ?
??
An example: 
? 1. Add the matrices 
? First, note that each matrix 
has dimensions of 3X3, so 
we are able to perform the 
addition. The result is shown 
at right:
? Solution: Adding 
corresponding entries we 
have
4 3 1 1 2 3
0 5 2 6 7 9
5 6 0 0 4 8
??
? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
??
? ? ? ?
3 1 4
6 2 7
5 10 8
?
??
??
?
??
??
?
??
Subtraction of matrices
? Now, we will subtract 
the same two matrices
? Subtract corresponding 
entries as follows:
4 3 1 1 2 3
0 5 2 6 7 9
5 6 0 0 4 8
??
? ? ? ?
? ? ? ?
? ? ?
? ? ? ?
? ? ? ?
??
? ? ? ?
4 ( 1) 3 2 1 3
0 6 5 ( 7) 2 9
5 0 6 ( 4) 0 8
? ? ? ? ?
??
??
? ? ? ? ?
??
?? ? ? ? ? ?
??
5 5 2
6 12 11
5 2 8
??
??
??
??
??
??
??
??
= 
Scalar Multiplication
? The scalar product of a number k and a 
matrix A is the matrix denoted by kA, 
obtained by multiplying each entry of A by 
the number k . The number k is called a 
scalar. In mathematical notation, 
ij
A k ka
??
?
??