Question

# Define is subtraction of matrices commutative and associative?

Matrices
Define is subtraction of matrices commutative and associative?

2020-12-16
Step 1
Consider three matrix A, B and C respectively.
$$A=\begin{bmatrix}a & b \\c & d \end{bmatrix} , B=\begin{bmatrix}e & f \\ g & h \end{bmatrix}, C=\begin{bmatrix}i & j \\ k & l \end{bmatrix}$$
Step 2
To define whether subtraction of matrices is commutative or associative.
Step 3
Proof:
Subtraction of matrices is not commutative.
Since , $$A-B=\begin{bmatrix}a & b \\c & d \end{bmatrix}-\begin{bmatrix}e & f \\ g & h \end{bmatrix}$$
$$=\begin{bmatrix}a-e & b-f \\ c-g & d-h \end{bmatrix}$$
And, $$B-A=\begin{bmatrix}e & f \\ g & h \end{bmatrix}-\begin{bmatrix}a & b \\c & d \end{bmatrix}$$
$$=\begin{bmatrix}e-a & f-b \\ g-c & h-d \end{bmatrix}$$
Therefore substraction of matrices is not commutative.
Step 4
Subtraction of matrices is not associative.
Since , $$(A+B)-C=\left(\begin{bmatrix}a & b \\c & d \end{bmatrix}+\begin{bmatrix}e & f \\ g & h \end{bmatrix}\right) -\begin{bmatrix}i & j \\ k & l \end{bmatrix}$$
$$=\begin{bmatrix}a+e-i & b+f-j \\ c+g-k & d+h-l \end{bmatrix}$$
And , $$(A-B)+(B-C)=\left(\begin{bmatrix}a & b \\c & d \end{bmatrix}-\begin{bmatrix}i & j \\ k & l \end{bmatrix}\right) +\left(\begin{bmatrix}e & f \\ g & h \end{bmatrix}-\begin{bmatrix}i & j \\ k & l \end{bmatrix}\right)$$
$$=\begin{bmatrix}a+e-2i & b+f-2j \\ c+g-2k & d+h-2l \end{bmatrix}$$
So, it is also not associative.