Question

Find matrices $A$and $C$ in $M\left(\mathrm{ℝ}\right)$ such that $AC=0$ , but$CA\ne 0$ , where 0 is the zero matrix.

Short answer

The required matrices are$A=\left[\begin{array}{cc}a& 0\\ b& 0\end{array}\right]$ and $C=\left[\begin{array}{cc}0& 0\\ c& d\end{array}\right]$.

Step-by-step solution

Write the definition of Rings

A ring can be defined as a non-empty set $\mathrm{ℝ}$whose elements deal with basically two operations: addition and multiplication, where elements belong to the real number$\mathrm{ℝ}$ .

Property of Rings:

Let the two matrices be:

$A=\left[\begin{array}{cc}a& 0\\ b& 0\end{array}\right]C=\left[\begin{array}{cc}0& 0\\ c& d\end{array}\right]$

Clearly, the product $AC$ will be given as:

$\begin{array}{rcl}AC& =& \left[\begin{array}{cc}a& 0\\ b& 0\end{array}\right]\left[\begin{array}{cc}0& 0\\ c& d\end{array}\right]\\ & =& \left[\begin{array}{cccccc}0& +& 0& 0& +& 0\\ 0& +& 0& 0& +& 0\end{array}\right]\\ & =& \left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\end{array}$

Also, the product $CA$ will be:

$\begin{array}{rcl}CA& =& \left[\begin{array}{cc}0& 0\\ c& d\end{array}\right]\left[\begin{array}{cc}a& 0\\ b& 0\end{array}\right]\\ & =& \left[\begin{array}{cccccc}0& +& 0& 0& +& 0\\ ca& +& db& 0& +& 0\end{array}\right]\\ & =& \left[\begin{array}{cc}0& 0\\ ca+db& 0\end{array}\right]\end{array}$

Therefore, we have:

$AC=0$ but $CA\ne 0$

Hence, the required matrices are:

$A=\left[\begin{array}{cc}a& 0\\ b& 0\end{array}\right]C=\left[\begin{array}{cc}0& 0\\ c& d\end{array}\right]$