We motivated matrix multiplication via obtaining one system of linear equations from another. Let us focus for a moment on obtaining a single equation. From

adding negative three times the first equation to the second equation gives:

The actual variable symbols are not relevant so we can simply work with the coefficients and constants representing them as entries in a one by six matrix for each equation separately

Above on the left side there is multiplication of scalar with a matrix and addition of matrices. It is only natural that we define those two operations.

While the example above multiplied a constant with a one times five matrix any scalar can be multiplied with any matrix.

Bear in mind that multiplying the one times one matrix whose single entry is five is not possible i.e., the following:

is not defined. No $1\times 1$ matrix can be multiplied by any $3\times 5$ matrix.

Any two $m\times n$ matrices can be added.

Here is an example

As with the usual addition whenever two matrices can be added the addition is:

commutative

$A+B=B+C$

associative

$A+(B+C)=(A+B)+C$

Due to associativity omitting the brackets as with the real numbers is accepted notation. There is a $m\times n$ zero matrix denoted by ${𝟘}_{m\times n}$ whose all entries are all zeroes and for any $m\times n$ matrix $A$ we have

We will simply write $𝟘$ when the number of rows and columns can be deduced from the context.

For every matrix $A$ there is a matrix $B$ such that

Typically $B$, which of course is $-1A$, is denoted via $\mathrm{-A}$ and we simply write

We can perform a number of scalar (column) matrix multiplications and matrix additions as with the real numbers.