Example 10-1. Find the derivative and integral of the matrix A=
[
1 x
sin x e−^2 x
]
Solution Taking the derivative of each element one finds
dA
dx =
[
0 1
cos x − 2 e−^2 x
]
Taking the integral of each element one finds
∫
A dt =
[
x^12 x^2
−cos x −^12 e−^2 x
]
+C
where C= (cij) 2 × 2 is an arbitrary constant matrix.
The Dot or Inner Product
The dot or inner product of a n-dimensional row vector R and n-dimensional
column vector C, where
R= (r 1 , r 2 , r 3 ,... , rn) and C=col(c 1 , c 2 , c 3 ,... , cn)
is a single number written as the matrix product
RC = (r 1 , r 2 , r 3 ,... , rn)
c 1
c 2
c 3
..
.
cn
=r 1 c 1 +r 2 c 2 +r 3 c 3 +···+rncn=
∑n
m=1
rmcm (10.5)
representing the summation of the products of the mth row vector element with the
mth column vector element, as mvaries from 1 to n. In order to calculate an inner
product the row vector and column vector must have the same number of elements.
Matrix Multiplication
Let A= (aij)m×n denote an m×n matrix and let B = (bij)p×q denote a p×q
matrix. If the dimensions n, p have the proper size, then the matrix Acan be right-
multiplied by the matrix B to produce a new matrix C. This matrix product is
written C=AB and this matrix product can only occur when the matrices Aand B
have the proper dimensions. For the matrix product AB =Am×nBp×q to exist it is
required that the dimension pof B must equal the dimension nof Aand whenever
this condition is satisfied, then the matrices are said to satisfy the compatibility
condition for matrix multiplication to occur. If the column dimension of Adoes not
