Previous article, you learned that matrix are two dimensional representation of data other than augmented matrix from a system of linear equations. Matrix operations such as addition is possible because you can add two matrices $A$ and $B$ by simply adding their corresponding elements which will give a thrid matrix as a result.

### Condition To Add Two Matrices

You can add a matrix like ordinary numbers simply by adding each corresponding elements of two matrices. This is only possible if size of both matrices are same.

Also, the order of matrix addition is not important because addition has commutative property.

Let $A$ and $B$ be two matrices of same size.

$A_{2 \times 3} = \begin{bmatrix} a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\end{bmatrix}_{2 \times 3}$

$B_{2 \times 3} = \begin{bmatrix} b_{11} & b_{12} & b_{13}\\b_{21} & b_{22} & b_{23}\end{bmatrix}_{2 \times 3}$

Both matrices of same size and the order of addition does not matter, then

$C_{2 \times 3} = A_{2 \times 3} + B_{2 \times 3} = B_{2 \times 3} + A_{2 \times 3}$

$C_{2 \times 3} = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} & a_{13} + b_{13}\\a_{2 \times 3} + b_{21} & a_{22} + b_{22} & a_{23} + b_{23}\end{bmatrix}$

$C_{2 \times 3} = \begin{bmatrix} c_{11} & c_{12} & c_{13}\\c_{21} & c_{22} & c_{23}\end{bmatrix}$

The addition operation has certain fundamental properties that applies to all real numbers. Since, matrix addition is also a common addition, these fundamental mathematical properties applies to them as well.

Commutative Property

If A and B are two independent matrices of same size, then

$A + B = B + A$

Example

$A = \begin{bmatrix} 1 & 2 & 3\\6 & 5 & 4\end{bmatrix} \hspace{3ex} B = \begin{bmatrix} 2 & 1 & 5\\1 & 1 & 0\end{bmatrix}$
$C = A + B = \begin{bmatrix} 3 & 3 & 8\\7 & 6 & 4\end{bmatrix} = B + A$ 

Associative Property

If A, B and C are three matrices of same size then,

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

Example

$A = \begin{bmatrix} 1 & 2 & 3\\6 & 5 & 4\end{bmatrix} \hspace{3ex} B = \begin{bmatrix} 2 & 1 & 5\\1 & 1 & 0\end{bmatrix} \hspace{3ex} C = \begin{bmatrix} 1 & 1 & 1\\1 & 2 & 1\end{bmatrix}$
$(A + B) + C = \begin{bmatrix} 3 & 3 & 8\\7 & 6 & 4\end{bmatrix} + \begin{bmatrix} 1 & 1 & 1\\1 & 2 & 1\end{bmatrix}$

$(A + B) + C = \begin{bmatrix} 4 & 4 & 9\\8 & 8 & 5\end{bmatrix}$

Similarly,

$A + (B + C) = \begin{bmatrix} 1 & 2 & 3\\6 & 5 & 4\end{bmatrix} + \begin{bmatrix} 3 & 2 & 6\\2 & 3 & 1\end{bmatrix}$

$A + (B + C) = \begin{bmatrix} 4 & 4 & 9\\8 & 8 & 5\end{bmatrix}$

Identity Property

The identity element for addition (+) is $0$. You add $0$ to element $a + 0 = a$. The result is always $a$. Therefore, if we add matrix A to a zero matrix, the result is A matrix itself.

$A + O = A$

Example

$A = \begin{bmatrix} 1 & 2 & 3\\6 & 5 & 4\end{bmatrix} \hspace{3ex} O = \begin{bmatrix} 0 & 0 & 0\\0 & 0 & 0\end{bmatrix}$

$A + O = \begin{bmatrix} 1 & 2 & 3\\6 & 5 & 4\end{bmatrix}$

There are other intresting properties of matrix addition which we discuss in future posts.