Operations on Matrices

Matrix operations include the arithmetic operations of addition, subtraction, multiplication of matrices. Also, we can find the transpose and inverse of a matrix, which can also be included as operations on matrices. The matrix operations help in combining two or more matrices into a single matrix.

Addition of Matrices

The addition of matrices is one of the basic operations which is performed on matrices. Two or more matrices of the same order can be added by adding the corresponding elements of the matrices. If $A=[a_{ij}]$ and $B=[b_{ij}]$ are two matrices of the same order then the addition of matrices $A$ and $B$ is: $A+B=[a_{ij}]+[b_{ij}]=[a_{ij}+b_{ij}]$.

${\begin{bmatrix}a_{11}&a_{12}&.....&a_{1n}\\a_{21}&a_{22}&.....&a_{2n}\\a_{31}&a_{32}&.....&a_{3n}\\.&.&.....&.\\.&.&.....&.\\a_{m1}&a_{m2}&.....&a_{mn}\end{bmatrix}}_{m\times n}$$+$ ${\begin{bmatrix}b_{11}&b_{12}&.....&b_{1n}\\b_{21}&b_{22}&.....&b_{2n}\\b_{31}&b_{32}&.....&b_{3n}\\.&.&.....&.\\.&.&.....&.\\b_{m1}&b_{m2}&.....&b_{mn}\end{bmatrix}}_{m\times n}$$=$

${\begin{bmatrix}a_{11}+b_{11}&a_{12}+b_{12}&.....&a_{1n}+b_{1n}\\a_{21}+b_{21}&a_{22}+b_{22}&.....&a_{2n}+b_{2n}\\a_{31}+b_{31}&a_{32}+b_{32}&.....&a_{3n}+b_{3n}\\.&.&.....&.\\.&.&.....&.\\a_{m1}+b_{m1}&a_{m2}+b_{m2}&.....&a_{mn}+b_{mn}\end{bmatrix}}_{m\times n}$

Example:

$A={\begin{bmatrix}2&4&6\\8&10&12\\14&16&18\end{bmatrix}}$ $B={\begin{bmatrix}1&3&5\\7&9&11\\13&15&17\end{bmatrix}}$

$A+B={\begin{bmatrix}2+1&4+3&6+5\\8+7&10+9&12+11\\14+13&16+15&18+17\end{bmatrix}}={\begin{bmatrix}3&7&11\\15&19&23\\27&31&35\end{bmatrix}}$

Subtraction of Matrices

Two matrices of the same order can be subtracted by subtracting the corresponding elements of the matrices. If $A=[a_{ij}]$ and $B=[b_{ij}]$ are two matrices of the same order then the subtraction of matrices $A$ and $B$ is: $A-B=[a_{ij}]-[b_{ij}]=[a_{ij}-b_{ij}]$.

${\begin{bmatrix}a_{11}&a_{12}&.....&a_{1n}\\a_{21}&a_{22}&.....&a_{2n}\\a_{31}&a_{32}&.....&a_{3n}\\.&.&.....&.\\.&.&.....&.\\a_{m1}&a_{m2}&.....&a_{mn}\end{bmatrix}}_{m\times n}$$-$ ${\begin{bmatrix}b_{11}&b_{12}&.....&b_{1n}\\b_{21}&b_{22}&.....&b_{2n}\\b_{31}&b_{32}&.....&b_{3n}\\.&.&.....&.\\.&.&.....&.\\b_{m1}&b_{m2}&.....&b_{mn}\end{bmatrix}}_{m\times n}$$=$

${\begin{bmatrix}a_{11}-b_{11}&a_{12}-b_{12}&.....&a_{1n}-b_{1n}\\a_{21}-b_{21}&a_{22}-b_{22}&.....&a_{2n}-b_{2n}\\a_{31}-b_{31}&a_{32}-b_{32}&.....&a_{3n}-b_{3n}\\.&.&.....&.\\.&.&.....&.\\a_{m1}-b_{m1}&a_{m2}-b_{m2}&.....&a_{mn}-b_{mn}\end{bmatrix}}_{m\times n}$

Example:

$A={\begin{bmatrix}2&4&6\\8&10&12\\14&16&18\end{bmatrix}}$ $B={\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}}$

$A-B={\begin{bmatrix}2-1&4-2&6-3\\8-4&10-5&12-6\\14-7&16-8&18-9\end{bmatrix}}={\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}}$