# Matrix Addition & Subtraction Of Two Matrices

## Matrix Addition

Before going into matrix addition, let us have a brief idea what are matrix.

In mathematics, a matrix is a rectangular array of numbers, expression 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 like- Square matrix, Row matrix, Null matrix, Diagonal matrix, scalar matrix etc.

Let us now focus on how to perform the basic operation on matrices with matrix addition.

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 which define addition of matrix. They are as follows:

1.  Consider two matrices A & B. These matrices can be added iff(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 $3 \times 4$.
1. The addition of matrices is not defined for matrices of different sizes.

Let us take an Example for this:

Example- 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 order of matrix B is 3X2. So, matrix A and B cannot be added together.

Let us take another Example,

Let, 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 = $\begin{bmatrix} 5 & 9 & 10\cr 13 & 10 & 12\cr 14 & 13 & 15 \end{bmatrix}$

## Matrix Subtraction

Matrix subtraction is exactly 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 addition of the inverse of a matrix to the given matrix, i.e. if matrix Q have 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 = $\begin{bmatrix} a-j & b-k & c-l\cr d-m & e-m & f-o\cr g-p & h-q & i-r \end{bmatrix}$

and we know,

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 ‘n’ number of matrices to be added or subtracted.

Now that we are thoroughly through the process of matrix addition and subtraction to know more about operations on matrices, download BYJU’S-The learning app.