Question

We’ve shown that matrix multiplication is generally not commutative, meaning that as a general rule for two

matrices A and B, A ⋅ B ≠ B ⋅ A. Explain why F ⋅ G = G ⋅ F in each of the following examples.
a. F = ????1 3
2 0
???? , G = ????2 6
4 0
????
b. F = ????
1 3 2
3 1 2
4 3 2
???? , G = ????
0 0 0
0 0 0
0 0 0
????
c. F = ????
1 3 2
3 1 2
4 3 2
???? , G = ????
1 0 0
0 1 0
0 0 1
????
d. F = ????
1 3 2
3 1 2
4 3 2
???? , G = ????
3 0 0
0 3 0
0 0 3

Answer 1

a) Observe that

`G=\left[\begin{array}{ccc}2&6\\4&0\end{array}\right] =2\left[\begin{array}{ccc}1&3\\2&0\end{array}\right] =2F`

Then,
`FG=F(2F)=2FF, \text{ and } GF=(2F)F=2FF`

b) The zero matrix satisfies that for every matrix B such that the product is well defined,
`0B=0=B0.`

Since the matrix G is the zero matrix then
`FG=F0=0F=GF`

c) The identity(Id) matrix satisfies that for that for every matrix B such that the product is well defined Id*B=B=B*Id. Observe that G is the identity matrix, then FG=F*Id=F=Id*F=GF

d) Observe that

`G=\left[\begin{array}{ccc}3&0&0\\0&3&0\\0&0&3\end{array}\right] =3\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right] =3Id`.

Then
`GF=3Id*F=3F\\FG=F(3Id)=3F*Id=3F`