Problem 13.4
If[
ab
cd]
Y[
3 − 1
10]
=[
10
01]finda,b,c,d.We have
[
ab
cd
]
+[
3 − 1
10]
=[
a+ 3 b− 1
c+ 1 d+ 0]
=[
10
01]Equating corresponding elements gives
a+ 3 =1soa=− 2
b− 1 =0sob= 1
c+ 1 =0soc=− 1
d=1sod= 1Note that an equation in 2×2 matrices is equivalent tofourequations in terms of the
elements. Alternatively we can simply subtract the matrices:
[
ab
cd]
=[
10
01]
−[
3 − 1
10]
=[
− 21
− 11]I think you will agree that most of the above is fairly sensible and you might expect
that an obvious generalisation of addition of matrices would be to define multiplication
of matrices as multiplying corresponding elements. However, this is not the most useful
definition. As noted above, we define multiplication to fit in with the structure of linear
equations – ‘plugging rows into columns’. For this to be possible, without trying to fit
a 3-pin plug into a 2-pin socket, for example, we can only formABifAhas the same
number of columns asBhas rows. We then sayAandBareconformableand we form the
productABby multiplying each row ofAinto each column ofB, multiplying elements in
pairs and adding up the results. The formal definition looks more fearsome than it actually
is – have a look at Problem 13.5 to make sense of it.
Thus let:
A=[aij] (m×n)
B=[bij] (n×r)Then the elements of the product matrix:
C=AB=[cij] (m×r)are given by
cij=∑nk= 1aikbkj