Here you will learn how to add matrix and properties of addition of matrices with examples.
Let’s begin –
Addition of Matrices
Let A, B be two matrices, each of order \(m \times n\). Then their sum A + B is a matrix of order \(m \times n\) and is obtained by adding the correspoding elements of A and B.
Thus, if A = \([a_{ij}]_{m\times n}\) and B = \([b_{ij}]_{m\times n}\) are two matrices of the same order, their sum A + B is defined to be the matrix of order \(m\times n\) such that
\((A + B)_{ij}\) = \(a_{ij}\) + \(b_{ij}\) for i = 1, 2, ……. , m and j = 1, 2, ……. n
Note : The sum of two matrices is defined by only when they are of the same order.
Example : If A = \(\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}\), B = \(\begin{bmatrix} 6 & 5 & 4 \\ 3 & 2 & 1 \end{bmatrix}\), then
A + B = \(\begin{bmatrix} 1 + 6 & 2 + 5 & 3 + 4 \\ 4 + 3 & 5 + 2 & 6 + 1 \end{bmatrix}\) = \(\begin{bmatrix} 7 & 7 & 7 \\ 7 & 7 & 7 \end{bmatrix}\)
If A = \(\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}\), B = \(\begin{bmatrix} -1 & 2 & 1 \\ 3 & 2 & 1 \\ 2 & 5 & -2 \end{bmatrix}\), then A + B is not defined, because A and B are not of the same order.
Properties of Matrix Addition
(a) Commutativity : If A and B are two \(m\times n\) matrices, then A + B = B + A. i.e. matrix addition is commutative.
(b) Associativity : If A, B, C are three matrices of the same order, then (A + B) + C = A + (B + C) i.e. matrix addition is associative.
(c) Existence of Identity : The null matrix is the identity element for matrix addition.
(d) Existence of Inverse : for every matrix A = \([a_{ij}]_{m\times n}\) there exist a matrix \([-a_{ij}]_{m\times n}\), denoted by -A, such that A + (-A) = O = (-A) + A
(e) Cancellation Laws : If A, B, C are matrices of the same order, then
A + B = A + C \(\implies\) B = C
and, B + A = C + A \(\implies\) B = C
