Matrices. Operations with matrices.

A matrix of size $m\times n$ is called a rectangular table of numbers $a_{ij},,i=1,2,...,m,$ $j=1,2,...,n$,

$$A=\left(\begin{array}{cccc}{a}_{11}& a_{12}& ...& a_{1n}\\ a_{21}& a_{22}& ...& a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& ...& a_{mn}\end{array}\right)$$

consisting of $m$ rows and $n$ columns.

The sum $A+B$ of matrices of size $m\times n$, $A=a_{ij}$ and $B=b_{ij}$, is defined as the matrix $C=c_{ij}$ of the same order, where each element is equal to the sum of the corresponding elements of matrices $A$ and $B$:

$$A+B=\left(\begin{array}{cccc}{a}_{11}& a_{12}& ...& a_{1n}\\ a_{21}& a_{22}& ...& a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& ...& a_{mn}\end{array}\right)+\left(\begin{array}{cccc}{b}_{11}& b_{12}& ...& b_{1n}\\ b_{21}& b_{22}& ...& b_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ b_{m1}& b_{m2}& ...& b_{mn}\end{array}\right)=$$

$$=\left(\begin{array}{cccc}{a}_{11}+b_{11}& a_{12}+b_{12}& ...& a_{1n}+b_{1n}\\ a_{21}+b_{21}& a_{22}+b_{22}& ...& a_{2n}+b_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}+b_{m1}& a_{m2}+b_{m2}& ...& a_{mn}+b_{mn}\end{array}\right)$$

The product $\alpha A$ of a matrix $A=a_{ij}$ by a number $\alpha$ (real or complex) is defined as the matrix $B=b_{ij}$, where each element is equal to the product of the number $\alpha$ by the corresponding element of matrix $A$:

$$\alpha A=\alpha \left(\begin{array}{}{a}_{11}& a_{12}& ...& a_{1n}\\ a_{21}& a_{22}& ...& a_{2n}\\ ⋮& ⋮& ⋮& ⋮\\ a_{m1}& a_{m2}& ...& a_{mn}\end{array}\right)=\left(\begin{array}{}\alpha a_{11}& \alpha a_{12}& ...& \alpha a_{1n}\\ \alpha a_{21}& \alpha a_{22}& ...& \alpha a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ \alpha a_{m1}& \alpha a_{m2}& ...& \alpha a_{mn}\end{array}\right).$$

The product $AB$ of a matrix $A=a_{ij}$ of size $m\times n$ by a matrix $B=b_{ij}$ of size $n\times k$ is defined as the matrix $C=c_{ij}$ of size $m\times k$, where each element standing in the $i$-th row and $j$-th column is equal to the sum of the products of the corresponding elements of the $i$-th row of matrix $A$ and the $j$-th column of matrix $B$:

$$A\times B=\left(\begin{array}{}{a}_{11}& a_{12}& ...& a_{1n}\\ a_{21}& a_{22}& ...& a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& ...& a_{mn}\end{array}\right)\times \left(\begin{array}{}{b}_{11}& b_{12}& ...& b_{1k}\\ b_{21}& b_{22}& ...& b_{2k}\\ ⋮& ⋮& \ddots & ⋮\\ b_{n1}& b_{n2}& ...& b_{nk}\end{array}\right)=$$ $$=\left(\begin{array}{}\sum _{\nu =1}^n{a}_{1\nu }{b}_{\nu 1}& \sum _{\nu =1}^n{a}_{1\nu }{b}_{\nu 2}& ...& \sum _{\nu =1}^n{a}_{1\nu }{b}_{\nu k}\\ \sum _{\nu =1}^n{a}_{2\nu }{b}_{\nu 1}& \sum _{\nu =1}^n{a}_{2\nu }{b}_{\nu 2}& ...& \sum _{\nu =1}^n{a}_{2\nu }{b}_{\nu k}\\ ⋮& ⋮& \ddots & ⋮\\ \sum _{\nu =1}^n{a}_{m\nu }{b}_{\nu 1}& \sum _{\nu =1}^n{a}_{m\nu }{b}_{\nu 2}& ...& \sum _{\nu =1}^n{a}_{m\nu }{b}_{\nu k}\end{array}\right)=C.$$

The matrix $A^T$ is called the transpose of the matrix $A$ if the condition $a_{ij}^T=a_{ji}$ is satisfied for all $i,j$, where $a_{ij}$ and $a_{ij}^T$ are the elements of matrices $A$ and $A^T$ respectively:

$$A=\left(\begin{array}{cccc}{a}_{11}& a_{12}& ...& a_{1n}\\ a_{21}& a_{22}& ...& a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& ...& a_{mn}\end{array}\right)\Rightarrow A^T=\left(\begin{array}{cccc}{a}_{11}& a_{21}& ...& a_{m1}\\ a_{12}& a_{22}& ...& a_{m2}\\ ⋮& ⋮& \ddots & ⋮\\ a_{1n}& a_{2n}& ...& a_{mn}\end{array}\right)$$

EXAMPLES.

1. Given

$$A=\left(\begin{array}{ccc}2& 1& 3\\ 5& 3& -6\\ -1& 2& 4\end{array}\right);B=\left(\begin{array}{ccc}6& -3& 2\\ 4& -2& 5\\ 3& 2& 7\end{array}\right).$$

Compute

а) $A+B;$

b) $3A;$

c) $AB;$

d) $A^T.$

Solution.

а) $$A+B=\left(\begin{array}{ccc}2& 1& 3\\ 5& 3& -6\\ -1& 2& 4\end{array}\right)+\left(\begin{array}{ccc}6& -3& 2\\ 4& -2& 5\\ 3& 2& 7\end{array}\right)=\left(\begin{array}{ccc}2+6& 1-3& 3+2\\ 5+4& 3-2& -6+5\\ -1+3& 2+2& 4+7\end{array}\right)=\left(\begin{array}{ccc}8& -2& 5\\ 9& 1& -1\\ 2& 4& 11\end{array}\right).$$

b) $$3A=3\left(\begin{array}{ccc}2& 1& 3\\ 5& 3& -6\\ -1& 2& 4\end{array}\right)=\left(\begin{array}{ccc}3\cdot 2& 3\cdot 1& 3\cdot 3\\ 3\cdot 5& 3\cdot 3& 3\cdot (-6)\\ 3\cdot (-1)& 3\cdot 2& 3\cdot 4\end{array}\right)=$$ $$=\left(\begin{array}{ccc}6& 3& 9\\ 15& 9& -18\\ -3& 6& 12\end{array}\right).$$

c) $$AB=\left(\begin{array}{ccc}2& 1& 3\\ 5& 3& -6\\ -1& 2& 4\end{array}\right)\left(\begin{array}{ccc}6& -3& 2\\ 4& -2& 5\\ 3& 2& 7\end{array}\right)=$$ $$\left(\begin{array}{ccc}2\cdot 6+1\cdot 4+3\cdot 3& 2\cdot (-3)+1\cdot (-2)+3\cdot 2& 2\cdot 2+1\cdot 5+3\cdot 7\\ 5\cdot 6+3\cdot 4-6\cdot 3& 5\cdot (-3)+3\cdot (-2)-6\cdot 2& 5\cdot 2+3\cdot 5-6\cdot 7\\ -1\cdot 6+2\cdot 4+4\cdot 3& -1\cdot (-3)+2\cdot (-2)+4\cdot 2& -1\cdot 2+2\cdot 5+4\cdot 7\end{array}\right)=$$ $$=\left(\begin{array}{ccc}25& -2& 30\\ 24& -33& -17\\ 14& 7& 36\end{array}\right).$$

d) $$A^T=\left(\begin{array}{ccc}2& 5& -1\\ 1& 3& 2\\ 3& -6& 4\end{array}\right).$$

Tags: matrix, operations with matrices