(i) By analogy with how we have rewritten the pairs of equations we ‘multiply’ by
plugging rows of the first matrix into columns of the second matrix in the product. So:
[
− 10
12
][
x
y
]
=
[
(− 1 )x+ 0 y
( 1 )x+ 2 y
]
=
[
−x
x+ 2 y
]
The quantities−x,x+ 2 yare the elements of the new 2×1 matrix on the right-
hand side.
(ii) [
23
41
][
2
1
]
=
[
2 × 2 + 3 × 1
4 × 2 + 1 × 1
]
=
[
7
9
]
(iii) In this case the second factor has two columns. No problem, the pattern is
[
3 − 1
42
][
12
− 11
]
=
[
3 × 1 +(− 1 )(− 1 ) 3 × 2 +(− 1 )( 1 )
4 × 1 + 2 (− 1 ) 4 × 2 + 2 × 1
]
=
[
45
210
]
a2×2 matrix. Essentially we are treating each of the two columns in the second
factor as we did the single columns in (i) and (ii). Take care with the signs in these
calculations!
It is common to use the notationaijfor the elements of a matrix, the subscriptsi,j,
denoting theith row andjth column. We now give the general definition of a matrix.
AnmxnmatrixAis an array ofmnnumbers withmrows andncolumns denoted by:
A=
a 11 a 12 ... a 1 n
a 21 a 22 a 2 n
..
.
..
.
am 1 am 2 ... amn
The number
row column
aij is theijth element ofA,readas‘a,i,j’
We sometimes write
A=[aij]
The case when the number of rows and columns is equal, i.e.m=n, is very important.
Such matrices are calledsquare, and we will have a lot to say about them later on.
A matrix does not have a numerical value (whereas adeterminant–seeSection13.4–
does). Note that round brackets are sometimes used for matrices. This is not advised since
when handwritten such brackets can be confused with determinant lines.
We will see more on the algebra of matrices, but for now just note that:
Addition of matrices isassociative:
A+(B+C)=(A+B)+C
andcommutative:
A+B=B+A