1. ## Matrix

Matrix is defined by multiplication and addition operations.
Why is not matrix defined by subtraction...
Why is it meanless....?

2. ## Re: Matrix

You can subtract 2 matrices. If you have a problem visualizing it, just multiply one of the matrices by -1 and add them together, it's the same result.
Are you thinking matrix division?

3. ## Re: Matrix

You don't mean "Matrix is defined by multiplication and addition operations." (emphasis mine)

You mean that
"multiplication and addition operations are defined for matrices".

You don't need to define subtraction separately for matrices (or other algebraic entities) because "subtraction" is just adding the additive inverse.

Multiplication of a matrix by a number (scalar) is defined and the additive inverse of a matrix is just that matrix multiplied by -1.

To "subtract" the matrix $\displaystyle \begin{bmatrix}1 & 3 \\ 2 & 2\end{bmatrix}$ from the matrix $\displaystyle \begin{bmatrix}3 & 2 \\ 4 & 6\end{bmatrix}$, add the additive inverse, $\displaystyle -1\begin{bmatrix}1 & 3 \\ 2 & 2\end{bmatrix}= \begin{bmatrix}-1 & -3 \\ -2 & -2\end{bmatrix}$.

$\displaystyle \begin{bmatrix}3 & 2 \\ 4 & 6\end{bmatrix}- \begin{bmatrix}1 & 3 \\ 2 & 2\end{bmatrix}=$$\displaystyle \begin{bmatrix}3 & 2 \\ 4 & 6\end{bmatrix}+ \begin{bmatrix}-1 & -3 \\ -2 & -2\end{bmatrix}=$$\displaystyle \begin{bmatrix}2 & -1 \\ 2 & 4\end{bmatrix}$.