Understanding Engineering Mathematics

Multiplication of matrices, considered in Section 13.3, isassociative:

A(BC)=(AB)C

butnot commutative:

AB=BA

We also define some special matrices:

Zeroornull matrix– all elements zero. It satisfies the obvious rulesA+ 0 =A, 0 A=
0 ,etc.
Unit matrix, I– all elements zero except for one’s on the leading diagonal. It must
therefore be a square matrix, and it satisfies

IA=AI′=A(I′andImay be different size)

for any matrixA.
Inverse matrix–ifAis a square matrix then it may be possible to find another matrix
A−^1 such that:

AA−^1 =A−^1 A=I

Such a matrix is called theinverseofA. We show how to evaluate the inverse matrix in
Section 13.6.
Transpose– the transpose of a matrixA, denotedATis the matrix obtained by changing
the rows ofAinto columns, thus, ifAism×n,ATisn×m.

Exercise on 13.2

Write the following system of equations in matrix formAx=b. In each case, specify the
location of the coefficient−3inthematrixAin the formaij

(i) x− 3 y= 1 (ii) 3x+ 2 y−z= 1

2 x+ 4 y= 4 −x− 3 y+ 2 z= 1

x+y+z= 2

Answer

(i)

[

1 − 3

24

][

x

y

]

=

[

1

4

]

a 12 =− 3

(ii)

[ 32 − 1

− 1 − 32

111

][x

y

z

]

=

[ 1

1

2

]

a 22 =− 3

13.3 Adding and multiplying matrices

Two matrices areequalif and only if corresponding elements are identical. Formally, two
matrices are equal if and only if they are of the same size and the corresponding elements