Question

Let A be a$n\times n$ matrix and let 0 be the$n\times n$ matrix all of whose entries are zero. Show that the following are true.

a)$A\odot 0=0\odot A=0$

b)$A\vee 0=0\vee A=A$

c)$A\wedge 0=0\wedge A=0$

Short answer

$$\begin{gathered} A\odot 0=0\odot A=0 \\ A\vee 0=0\vee A=A \\ A\wedge 0=0\wedge A=0 \end{gathered}$$

Step-by-step solution

Step: 1

a) Let $A\odot 0=\left[b_{ij}\right]$.

Then$b{i}_j=(a_{i1}\wedge 0)\vee ...\vee (a_{ip}\wedge 0)=0$

Hence $A\odot 0=0$. Similarly$0\odot A=0$

Step: 2

b)$A\vee 0=[{a_i}_j\vee 0]={a_i}_j=A$

Hence $A\vee 0=A$. Similarly$0\vee A=A$

Step: 3

c)$A\wedge 0=[aij\wedge 0]=\left[0\right]=0$

Hence $A\wedge 0=0$. Similarly$0\wedge A=0$