MATH SOLVE

4 months ago

Q:
# We’ve shown that matrix multiplication is generally not commutative, meaning that as a general rule for twomatrices A and B, A ⋅ B ≠ B ⋅ A. Explain why F ⋅ G = G ⋅ F in each of the following examples.a. F = ????1 32 0???? , G = ????2 64 0????b. F = ????1 3 23 1 24 3 2???? , G = ????0 0 00 0 00 0 0????c. F = ????1 3 23 1 24 3 2???? , G = ????1 0 00 1 00 0 1????d. F = ????1 3 23 1 24 3 2???? , G = ????3 0 00 3 00 0 3

Accepted Solution

A:

Answer:a) Observe that[tex]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[/tex]Then, [tex]FG=F(2F)=2FF, \text{ and } GF=(2F)F=2FF[/tex]b) The zero matrix satisfies that for every matrix B such that the product is well defined, [tex]0B=0=B0.[/tex]Since the matrix G is the zero matrix then [tex]FG=F0=0F=GF[/tex]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=GFd) Observe that [tex]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[/tex].Then [tex]GF=3Id*F=3F\\FG=F(3Id)=3F*Id=3F[/tex]