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 = 3
2 0
, g = 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

a) Observe that

Then,

b) The zero matrix satisfies that for every matrix B such that the product is well defined,

Since the matrix G is the zero matrix then

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

.

Then

(3x-8) or (3x+8)

step-by-step explanation:

< xcy ≅ < xby

step-by-step explanation:

in δcax and δbax,

so, δcax ≅ δbax (side-angle-side or sas)

since δcax ≅ δbax, we can conclude < acx ≅ < abx

in δacy and δaby,

so, δacy ≅ δaby (side-angle-side or sas)

since δacy ≅ δaby, we can conclude < acy ≅ < aby

now,

< acy ≅ < aby

=> < acx + < xcy ≅ < abx + < xby

=> < acx + < xcy ≅ < acx + < xby   (since < acx ≅ < abx already proved above)

subtracting < acx from both sides, we get

< acx + < xcy -< acx ≅ < acx + < xby -< acx

cancelling out < acx and -< acx from both sides, we get

< xcy ≅ < xby (proved)