The document discusses the differences between solving matrix equations of the form AX = B and XA = B. It shows that when solving AX = B, X is obtained by multiplying the inverse of A with B. However, when solving XA = B, X is obtained by multiplying B with the inverse of A. An example is provided to demonstrate that the solutions are not equal, proving that (AX = B) ≠ (XA = B).

1. Name : Meutiah Nahrisyah
Class : C (ICP 2017)
( 𝑨𝑿 = 𝑩) ≠ (𝑿𝑨 = 𝑩)
If the matrix equation is 𝐴𝑋 = 𝐵, then
𝐴𝑋 = 𝐵
↔ 𝐴−1
𝐴𝑋 = 𝐴−1
𝐵 (both side multiplied from the left by 𝐴−1
)
↔ 1𝑋 = 𝐴−1
𝐵 (𝐴−1
. 𝐴 = 1)
↔ 𝑋 = 𝐴−1
𝐵
Thus, 𝑋 = 𝐴−1
𝐵 is obtained.
It’s different, if we want to solve the matrix equation in the from of 𝑋𝐴 = 𝐵, then
𝑋𝐴 = 𝐵
↔ 𝑋𝐴. 𝐴−1
= 𝐵𝐴−1
(both side multiplied from the right by 𝐴−1
)
↔ 𝑋1 = 𝐵𝐴−1
(𝐴. 𝐴−1
= 1
↔ 𝑋 = 𝐵𝐴−1
Thus, 𝑋 = 𝐵𝐴−1
is obtained.
Therefore, ( 𝑨𝑿 = 𝑩) ≠ (𝑿𝑨 = 𝑩)
Example :
Let 𝐴 = [
6 1
3 5
] 𝐴−1
= [
5 −1
−3 6
]
𝐵 = [
4 −3
2 5
]
Find the 𝑋 that satisfies the following equation :
a. 𝐴𝑋 = 𝐵
b. 𝑋𝐴 = 𝐵

2. Answer :
a. 𝐴𝑋 = 𝐵
𝑋 = 𝐴−1
𝐵
𝑋 = [
5 −1
−3 6
] × [
4 −3
2 5
]
𝑋 = [
18 −20
0 39
]
b. 𝑋𝐴 = 𝐵
𝑋 = 𝐵𝐴−1
𝑋 = [
4 −3
2 5
] × [
5 −1
−3 6
]
𝑋 = [
29 −22
− − 5 28
]
It is proven that ( 𝑨𝑿 = 𝑩) ≠ (𝑿𝑨 = 𝑩).