Chapter 3.5, Problem 32HP

To determine

To prove: The matrix addition is associative for any $2\times 2$ matrices.

Answer to Problem 32HP

Since, both left hand side and right hand side are proved equal, therefore, matrix addition is associative for $2\times 2$ matrices.

Explanation of Solution

Given information :

  $A=\left[\begin{array}{cc}-3& -4\\ 8& 6\end{array}\right],B=\left[\begin{array}{cc}5& -1\\ 2& -4\end{array}\right],\text{and }3A-4B+6C=\left[\begin{array}{cc}13& 22\\ 10& 4\end{array}\right]$ .

Calculation :

It is required to prove that matrix addition is associative for $2\times 2$ . matrices.

Consider three matrices of the given order.

  $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],B=\left[\begin{array}{cc}e& f\\ g& h\end{array}\right]\text{and}C=\left[\begin{array}{cc}p& q\\ r& s\end{array}\right]$

To prove that $A+(B+C)=(A+B)+C$

For this, start with the left hand side

  $\begin{array}{l}A+(B+C)=\left[\begin{array}{ll}a\hfill & b\hfill \\ c\hfill & d\hfill \end{array}\right]+\left(\left[\begin{array}{ll}e\hfill & f\hfill \\ g\hfill & h\hfill \end{array}\right]+\left[\begin{array}{ll}p\hfill & q\hfill \\ r\hfill & s\hfill \end{array}\right]\right)\\ =\left[\begin{array}{ll}a\hfill & b\hfill \\ c\hfill & d\hfill \end{array}\right]+\left[\begin{array}{ll}e+p\hfill & f+q\hfill \\ g+r\hfill & h+s\hfill \end{array}\right]\\ =\left[\begin{array}{ll}a+e+p\hfill & b+f+p\hfill \\ c+g+r\hfill & d+h+s\hfill \end{array}\right]\\ =\left[\begin{array}{ll}a+e\hfill & b+f\hfill \\ c+g\hfill & d+h\hfill \end{array}\right]+\left[\begin{array}{ll}p\hfill & q\hfill \\ r\hfill & s\hfill \end{array}\right]\end{array}$

It can be done since addition is associative.

  $\begin{array}{l}A+(B+C)=\left(\left(\left[\begin{array}{ll}a\hfill & b\hfill \\ c\hfill & d\hfill \end{array}\right]+\left[\begin{array}{ll}e\hfill & f\hfill \\ g\hfill & h\hfill \end{array}\right]\right)+\left[\begin{array}{ll}p\hfill & q\hfill \\ r\hfill & s\hfill \end{array}\right]\right)\\ =(A+B)+C\end{array}$

Since, both left hand side and right hand side are proved equal, therefore, matrix addition is

Associative for $2\times 2$ matrices.