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Unformatted text preview: e i th column (for i = 1, 2,
.. , , m), t he r esulting m atrix is called t he t ranspose of A and is d enoted by AT.
I t is evident t hat A T is a n n x m m atrix. For example, if ~ ~ : : :; . .. : :: : :::; . ..•••.. : :: : : : ; + b mi ) (a m 2 + b m 2) .. , ( a mn I 37 + b mn ) or
A +B = ( aij + b ij)mxn Note t hat two matrices can b e a dded only if t hey a re of t he same order. 2. Multiplication o f a Matrix by a Scalar
We define t he m ultiplication of a m atrix A by a scalar c as
t hen AT = [ 21 32 1
cA = c Thus, if c ami ca m 2 t hen
(B.54) Note t hat 3. Matrix Multiplication
We define t he p roduct
A B=C (B.55) B.6-2 in which C ij, t he e lement of C in t he i th row a nd j th column, is found by adding
t he p roducts of t he elements of A in t he i th row with t he corresponding elements
of B in t he j th column. Thus, Matrix Algebra We shall now define matrix operations, such as addition, subtraction, multiplication, a nd division of matrices. T he definitions should be formulated so t hat t hey
are useful in t he m anipulation of matrices. n (B.56) = L aikhj
k =i T his result is shown below. 1. Addition o f M atrices
For two m atrices A a nd B , b oth o f t he same order (m x n), C ij a nd A= B= bnJ· we define t he s um A +B as '-----v------'~
A (mxn) ~ ~~ ~
-----v----- C (mxp) 38 Background Note carefully t hat t he number of columns of A must be equal to the number of
rows of B if this procedure is t o work. I n o ther words, A B, the product of matrices
A a nd B , is defined only if the number of columns of A is equal to the number of
rows of B . I f t his condition is not satisfied, the product A B is not defined and is
meaningless. W hen t he number of columns of A is equal to the number of rows of
B , m atrix A is s aid to be c onformable t o matrix B for the product A B. Observe
t hat if A is an m X n m atrix and B is a n n X p matrix, A and B are conformable
for t he product, a nd C is an m X p matrix.
We d emonstrate the use of t he rule in Eq. (B. 56) with the following examples. B.6 Vectors and Matrices 39 (B.60) A I==IA==A O f course, we m ust make sure t hat t he order of I is such t hat t he matrices a re
conformable for t he corresponding product.
4. Multiplication o f a Matrix by a Vector
Consider t he m atrix Eq. (B.52), which represents Eq. (B.4S). T he r ight-hand
o f Eq. (B.52) is a p roduct o f t he m x n m atrix A a nd a vector x. If, for t he
t ime being, we t reat t he vector x as if i t were a n n x 1 m atrix, then t he p roduct
A x, according t o t he m atrix multiplication rule, yields t he right-hand side of Eq.
(B.4S). Thus, we may multiply a m atrix by a vector by treating t he vector as if i t
were a n n x 1 m atrix. Note t hat t he c onstraint of conformability still applies. Thus,
in this case, x A is n ot defined and is meaningless.
sid~ 5. Matrix Inversion
To define t he inverse of a matrix, let us consider t he s et of equations
Yl a 12 al n Xl Y2 In b oth cases above, the two matrices are conformable. However, if we interchange
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