Matrix addition explains the addition of two or more matrices. Unlike arithmetic addition of numbers, matrix addition will follow different rules. The order of matrices should be the same, before adding them. Before going into the addition of the matrix, let us have a brief idea of what are matrices. In mathematics, a matrix is a rectangular array of numbers, expressions or symbols, arranged in rows and columns. Horizontal Rows are denoted by “m” whereas the Vertical Columns are denoted by “n.” Thus a matrix (m x n) has m and n numbers of rows and columns respectively. We also know about different types of matrices such as square matrix, row matrix, null matrix, diagonal matrix, scalar matrix, identity matrix, diagonal matrix, triangular matrix, etc. Now, let us now focus on how to perform the basic operation on matrices such as matrix addition and subtraction with examples.

What is Matrix Addition?

Addition of matrix is the basic operation performed, to add two or more matrices. Matrix addition is possible only if the order of the given matrices are the same. By order we mean, the number of rows and columns are the same for the matrices. Hence, we can add the corresponding elements of the matrices. But if the order is different then matrix addition is not possible. Suppose A = [a_ij]_mxn and B = [b_ij]_mxn are two matrices of order m x n, then the addition of A and B is given by;

A + B = [a_ij]_mxn + [b_ij]_mxn = [a_ij + b_ij]_mxn

By recalling the small concept of addition of algebraic expressions, we know that while the addition of algebraic expressions can only be done with the corresponding like terms, similarly the addition of two matrices can be done by addition of corresponding terms in the matrix.

There are basically two criteria that define the addition of a matrix. They are as follows:

1.  Consider two matrices A & B. These matrices can be added if (if and only if) the order of the matrices are equal, i.e. the two matrices have the same number of rows and columns. For example, say matrix A is of the order \(3 \times 4\), then the matrix B can be added to matrix A if the order of B is also \(3 \times 4\).
2. The addition of matrices is not defined for matrices of different sizes.

Properties of Matrix Addition

The basic properties of matrix addition are similar to the addition of real numbers. Go through the properties given below:

Assume that, A, B and C be three m x n matrices, The following properties hold true for the matrix addition operation. 

• Commutative Property: If A and B are two matrices of the same order, say m x n, then the addition of the two matrices is commutative, i.e., A + B = B + A
• Associative Property: If A, B and C are three matrices of the same order, say m x n, then the addition of the three matrices is associative, i.e., A + (B + C) = (A + B) + C
• Additive identity: For any m x n matrix, there is an identity element. Thus, if A is m x n order matrix, then the additive identity of A will be zero matrix of same order, such that, A + O =  A ( where O is an additive identity)
• Additive inverse: If A is any matrix of order m x n, then the additive inverse of A will be B (= -A) of same order, such that, A + B = O. Hence, the sum of matrix and its additive inverse results in a zero matrix

What is Matrix Subtraction?

Matrix subtraction is exactly the same as matrix addition. All the constraints valid for addition are also valid for matrix subtraction. Matrix subtraction can only be done when the two matrices are of the same size. Subtraction cannot be defined for matrices of different sizes. Mathematically,
\( P – Q = P + (-Q) \)
In other words, it can be said that matrix subtraction is an addition of the inverse of a matrix to the given matrix, i.e. if matrix Q has to be subtracted from matrix P, then we will take the inverse of matrix Q and add it to matrix P.

Let, P = \(\begin{bmatrix}
a & b & c\cr
d & e & f \cr
g & h & i
\end{bmatrix}\) and \( Q = \begin{bmatrix}
j & k & l\cr
m & n & 0\cr
p & q & r
\end{bmatrix}\)

So, P-Q = \(\begin{bmatrix}
a-j & b-k & c-l\cr
d-m & e-m & f-o\cr
g-p & h-q & i-r
\end{bmatrix}\)

As we know, the matrix addition and subtraction undergoes the same process, the matix addition of the given array of elements are written as follows:

P+Q = \(\begin{bmatrix}
a+j & b+k & c+l\cr
d+m & e+n & f+o\cr
g+p & h+q & i+r
\end{bmatrix}\)

The main concept behind the addition or subtraction of two matrices is the addition or subtraction of corresponding terms of the given matrix.

Similarly, the given method can be generalized for the ‘n’ number of matrices to be added or subtracted.

Solved Examples on Matrix Addition

Example 1: Addition of matrices with different order.

Let, A = \(\begin{bmatrix}
4 & 7\cr
3 & 2
\end{bmatrix} \) and B = \(\begin{bmatrix}
1 & 2 & 3\cr
5 & 7 & 9
\end{bmatrix}\)

A+B matrix cannot be defined as the order of matrix A is 2×2 and the order of matrix B is 3X2. So, matrices A and B cannot be added together.

Example 2: Addition of matrices with the same order.

Let us add two 3 x 3 matrices.

Suppose, P =\(\begin{bmatrix}
2 & 4 & 3\cr
5 & 7 & 8 \cr
9 & 6 & 7
\end{bmatrix} \) and Q =\( \begin{bmatrix}
3 & 5 & 7\cr
8 & 3 & 4\cr
5 & 7 & 8
\end{bmatrix}\)

P+Q matrix can be found out by adding elements of P to the corresponding elements of Q.  So, value of matrix P+Q is
P + Q =  \(\begin{bmatrix} 2+3 & 4+5 & 3+7 \\ 5+8 & 7+3 & 8+4 \\ 9+5 & 6+7 & 7+8 \end{bmatrix}\)

P + Q = \(\begin{bmatrix}
5 & 9 & 10\cr
13 & 10 & 12\cr
14 & 13 & 15
\end{bmatrix}\)

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