Question

Give an example of two non zero 2 × 2 matrices a, b such that ab = 0.

Answer 1

a= |0 0| and b= |0 0|
-----|0 1| ----------- |1 0|

Here a and b both is non zero. But a×b = b×a = 0

Answer 2

The non zero matrix - a matrix with non zero elements is called the non zero matrix.

Let us consider,

2 × 2 matrix  a

$a = \left[\begin{array}{ccc}2&1\\4&2\end{array}\right]$

2 × 2 matrix  b

$b = \left[\begin{array}{ccc}-1&-2\\2&4\end{array}\right]$

The product of matrices ab is given by,

$ab = \begin{pmatrix}2&1\\ 4&2\end{pmatrix}\begin{pmatrix}-1&-2\\ 2&4\end{pmatrix}$

$ab=\begin{pmatrix}2\left(-1\right)+1\cdot \:2&2\left(-2\right)+1\cdot \:4\\ 4\left(-1\right)+2\cdot \:2&4\left(-2\right)+2\cdot \:4\end{pmatrix}$

$ab=\begin{pmatrix}-2+2&-4+4\\ -4+4&-8+8 \end{pmatrix}$

$\therefore ab =\begin{pmatrix}0&0\\ 0&0\end{pmatrix}$