Page 208 - Advanced engineering mathematics

188 CHAPTER 7 Matrices and Linear Systems

There are three operations we will define for matrices: addition, multiplication by a real or
complex number, and multiplication. These are defined as follows.
Addition of Matrices

If A=[a ij ] and B=[b ij ] are both n × m matrices, then their sum is defined to be the n × m
matrix A + B =[a ij + b ij ].

We add two matrices of the same dimensions by adding objects in the same locations in the
matrices. For example,

1 2 −3 −1 6 3 0 8 0
+ = .
4 0 2 8 12 14 12 12 16
We can think of this as adding respective row vectors, or respective column vectors, of the matrix.
Multiplication by a Scalar

Multiply a matrix by a scalar quantity (say a number or function) by multiplying each
matrix element by the scalar. If A =[a ij ], then cA =[ca ij ]. For example,
√
⎛ ⎞ ⎛ ⎞
−3 −3 2
√
√ ⎜ 4 ⎟ ⎜ 4 2 ⎟
2 ⎜ ⎟ = ⎜ √ ⎟ .
⎝ 2t ⎠ ⎝ 2t 2 ⎠
√
sin(2t) 2sin(2t)

This is the same as multiplying each row vector, or each column vector, by c. As another
example,
t t
2 e 2cos(t) e cos(t)
cos(t) = .
sin(t) 4 cos(t)sin(t) 4cos(t)
Multiplication of Matrices

Let A =[a ij ] be n × k and B =[b ij ] be k × m. Then the product AB is the n × m matrix
whose i, j element is
a i1 b 1 j + a i2 b 2 j + ··· + a ik b kj ,

or
k

a is b sj .
s=1

k
This is the dot product of row i of A with column j of B (both are vectors in R ):
i, j element of AB = ( row i of A) · ( column j of B)
= (a i1 ,a i2 ,··· ,a ik ) · (b 1 j ,b 2 j ,··· ,b kj )
= a i1 b 1 j + a i2 b 2 j + ··· + a ik b kj .

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October 14, 2010 14:23 THM/NEIL Page-188 27410_07_ch07_p187-246