JEE Main & Advanced Mathematics Determinants & Matrices Addition and Subtraction of Matrices

If \[A={{[{{a}_{ij}}]}_{m\times n}}\]and \[B={{[{{b}_{ij}}]}_{m\times n}}\]are two matrices of the same order then their sum \[A+B\] is a matrix whose each element is the sum of corresponding elements i.e., \[A+B={{[{{a}_{ij}}+{{b}_{ij}}]}_{m\times n}}\].

Similarly, their subtraction \[A-B\] is defined as

\[A-B={{[{{a}_{ij}}-{{b}_{ij}}]}_{m\times n}}\]

Matrix addition and subtraction can be possible only when matrices are of the same order.

Properties of matrix addition : If A, B and C are matrices of same order, then

(i) \[A+B=B+A\]                    (Commutative law)                 

(ii) \[(A+B)+C=A+(B+C)\]    (Associative law)

(iii) \[A+O=O+A=A,\]where O is zero matrix which is additive identity of the matrix.

(iv) \[A+(-A)=0=(-A)+A\], where \[(-A)\] is obtained by changing the sign of every element of A, which is additive inverse of the matrix.

(v) \[\left. \begin{align}  & A+B=A+C \\  & B+A=C+A \\  \end{align} \right\}\Rightarrow B=C\]          (Cancellation law)