Properties of Matrices
1. Two matrices A= (aij)m×nand B= (bij)m×nhaving the same dimension are equal
if aij =bij for all values of iand j. Equality is expressed A=B.
2. Two matrices A= (aij)m×n and B= (bij)m×n of the same size can be added or
subtracted and the resulting matrices are denoted
C=A+B where C= (cij)m×nwith cij =aij +bij
D=A−B where D= (dij)m×nwith dij =aij −bij
Here like elements are added or subtracted.
3. If the matrix A= (aij)m×nis multiplied by a scalar β, the resulting matrix is
β A = (β a ij)m×n
That is, each component aij of Ais multiplied by the scalar β.
4. Matrices of the same size obey the following laws.
A+B=B+A commutative law
A+ (B+C) =(A+B) + C associative law
For αand βscalar quantities one can write the
scalar distributive laws
α(A+B) = αA +αB
(α+β)A= αA +βA
α(βA ) = (αβ )A
5. The zero matrix has all zeros for elements and can be expressed in one of the
forms [0]m×nor [0] or 0 or ̃ 0.
6. If the elements of the matrix Aare functions of a single variable, say t, one can
write aij =aij(t)or A=A(t) = (aij(t)) to emphasize this fact, then the derivative
of the matrix Ais given by
dA
dt =
(
da ij
dt
)
(10.3)
and the integral of the matrix Ais
∫
A(t)dt =
(∫
aij(t)dt
)
+C (10.4)
where Cis a constant matrix of appropriate size. Here the derivative of a matrix
is obtained by differentiating each element of the matrix and the integral of
the matrix is obtained by integrating each element within the matrix and the
constants of integration are collected into a constant matrix.
