Matrix Addition and Subtraction

Just like vectors, you can add and subtract matrices. The process is similar.

For both of these operations, the matrices must be of the same size. That is, if we want to add two matrices $A$ and $B$, they both have to be of size $m \times n$. The same is true for subtraction.

Addition

The way addition is defined for two matrices of size $m \times n$ is as follows:

For each row $i$ and each column $j$:

That means that you must add each element at position $ij$ of matrix $A$ with the corresponding element at position $ij$ of matrix $B$. That gives you the element of the resulting matrix at position $ij$.

Another way of expressing this, which might make it more clear:

Example

Here is a concrete numerical example.

Take two matrices

and

We compute

Subtraction

Subtraction follows a similar principle, where for two matrices of size $m \times n$:

For each row $i$ and column $j$:

or

Example

Take two matrices

and

We compute

Properties of Matrix Addition

Because of its definition in terms of scalar addition, matrix addition inherits its algebraic properties.

Commutative Property

For two matrices $A$ and $B$, the order of the operands does not make a difference:

Associative Property

For three matrices $A$, $B$, and $C$:

Closure Property

If $A$ and $B$ are both $m \times n$ matrices, then $A + B$ is also an $m \times n$ matrix.

Additive Identity

For some matrix $A$ of size $m \times n$, if $0_{m \times n}$ is the zero matrix (just like the zero vector, it is a matrix consisting of all $0$) of size $m \times n$:

Additive Inverse

If $A$ is a matrix of size $m \times n$:

Properties of Matrix Subtraction

Just like with matrix addition, matrix subtraction inherits all the properties from scalar subtraction. That means it’s non-commutative and non-associative. That means that generally,

and