Chapter 12.2, Problem 5LC

Solution

The type of multiplication, scalar or matrix, can help with a repeated matrix addition problem.

The scalar multiplication can help with a repeated matrix addition problem.

Given information: Matrices are, $A=\left[\begin{array}{cc}3& -1\\ 2& 0\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 3\\ -2& 2\end{array}\right]$ .

Formula used: Matrix multiplication is defined as,

  $\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]\times \left[\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right]=\left[\begin{array}{cc}{a}_{11}\times b_{11}+a_{12}\times b_{21}& a_{11}\times b_{12}+a_{12}\times b_{22}\\ a_{21}\times b_{11}+a_{22}\times b_{21}& a_{21}\times b_{12}+a_{22}\times b_{22}\end{array}\right]$ .

Scalar multiplication is defined as,

  $c\times \left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]=\left[\begin{array}{cc}c\times a_{11}& c\times a_{12}\\ c\times a_{21}& c\times a_{22}\end{array}\right]$

Calculation:

Let, matrix $A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]$ .

Then, addition of the matrix $A$ with itself ( $n$ times) can be written as,

  $\begin{array}{c}A+A+\dots +A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]+\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]+\dots +\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]\\ nA=n\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]\\ =\left[\begin{array}{cc}n{a}_{11}& n{a}_{12}\\ n{a}_{21}& n{a}_{22}\end{array}\right]\end{array}$

Therefore, the scalar multiplication can help with a repeated matrix addition problem.