Matrix Addition and Subtraction

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 Such as square matrix, row matrix, null matrix, diagonal matrix, scalar matrix, identity matrix, diagonal matrix, triangular matrix, etc.

What is 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 the 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 \(3 \times 4\).
  2. The addition of matrices is not defined for matrices of different sizes.

Example for Matrix Addition

Let us take an example for this:

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

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

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

Properties of Matrix Addition

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

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

  • A + B  = B + A    (commutative property)
  • A + (B + C)  = (A + B) + C    (associative property)
  • For any m x n matrix, there is an identity element) 

                  A + 0 =  A ( where 0 is an additive identity)

  • For any  m x n matrix A there is an m x n matrix B 

                 A + B =  O ( B is an additive inverse of A, which is equal to -A)

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

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

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

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.