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

Addition and Subtraction of Matrices

Category : JEE Main & Advanced

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)

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