Given an example of two matrices A and B such that A ≠ O, B ≠ O, AB = O and BA ≠ O.

Home » Given an example of two matrices A and B such that A ≠ O, B ≠ O, AB = O and BA ≠ O.

Given an example of two matrices A and B such that A ≠ O, B ≠ O, AB = O and BA ≠ O.

Solution:

We have $\boldsymbol{A} \neq \boldsymbol{O}, \boldsymbol{B} \neq \boldsymbol{O}, \boldsymbol{A B}=\boldsymbol{O}$ and $\boldsymbol{B A} \neq \boldsymbol{O}$ .
We need to find: $\boldsymbol{A}, \boldsymbol{B}$ .
Let’s take $\boldsymbol{A}=\left(\begin{array}{ll}\mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{1}\end{array}\right) \neq \boldsymbol{O}$ and $\boldsymbol{B}=\left(\begin{array}{cc}\mathbf{0} & \mathbf{1} \\ \mathbf{0} & \mathbf{0}\end{array}\right) \neq \boldsymbol{0}$ .
We have here $\boldsymbol{A B}=\left(\begin{array}{ll}\mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{1}\end{array}\right)\left(\begin{array}{ll}\mathbf{0} & \mathbf{1} \\ \mathbf{0} & \mathbf{0}\end{array}\right)=\left(\begin{array}{ll}\mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0}\end{array}\right)=\boldsymbol{0}$
And $B A=\left(\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right)\left(\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right)=\left(\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right) \neq 0$
Thus our required matrices are:
$A=\left(\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right)$ and $\quad B=\left(\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right)$