Page 514 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Appen. C.3 Rules of Matrix Operations 501

i.e.,
c^t — I x 2 + 2 x 0 + 2 x 3 — 8

It is evident that the number of columns in A must equal the number of rows in B, or
that the matrices must be conformable. We also note that A B ^ BA.
The postmultiplication of a matrix by a column matrix results in a column
matrix.
Example C.3-3

1 1 r iM I ^
1 5 2 3 = 20
2 1 3 h i I n

Premultiplication of a matrix by a row matrix (or transpose of a column matrix) results
in a row matrix.
Example C.3-4
1 1 1
[1 3 2] 1 5 2 = [8 18 13]
2 1 3

The transpose of a product A B = C is = B^A^.
Example C.3-5
1 1 2 1
Let A B =
2 3 1 1
■3 2' CT = ßT^T _ 2 1 1 2 ■3 7'
C = A B =
.7 5 1 1 1 3 .2 5.

Inversion of a matrix. Consider a set of equations
a , j Xj + «12 ^2 + ^13-^3
« 21-^1 + «22 ^2 "*■ ^23-^3 3^2 (C .3 -1)

031-^1 + «32 ^2 ^33^3 ^ 3^3
which can be expressed in the m atrix form
A X = Y (C .3-2)

Prem ultiplying by the inverse obtain the solution
X = A-^Y (C .3-3 )