Ask Questions, Get Answers

Want to ask us a question? Click here
Browse Questions
Home  >>  CBSE XII  >>  Math  >>  Matrices
0 votes

Show by an example that for $A\neq 0,B\neq 0,AB=0.$

Can you answer this question?

1 Answer

0 votes
  • If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B: $\begin{bmatrix}AB\end{bmatrix}_{i,j} = A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + A_{i,3}B_{3,j} ... A_{i,n}B_{n,j}$
To prove
AB=0 where $A\neq 0$ $B\neq 0$
AB=0 does not necessarily imply that A=0 or B=0 or both A=0 & B=0.Where 0 is a zero matrix.
For example
Let $A=\begin{bmatrix}1&-1\\-1 & 1\end{bmatrix}\neq 0$
$B=\begin{bmatrix}1&1\\1 & 1\end{bmatrix}\neq 0$
$AB=\begin{bmatrix}1 & -1\\-1 & 1\end{bmatrix}\begin{bmatrix}1&1\\1 & 1\end{bmatrix}$
$\;\;\;=\begin{bmatrix}1-1&1-1\\-1+1 & -1+1\end{bmatrix}$
$\;\;\;=\begin{bmatrix}0&0\\0 & 0\end{bmatrix}=0.$
Hence AB=0 but neither A=0 nor B=0.
answered Mar 23, 2013 by sharmaaparna1

Related questions

Ask Question
student study plans
JEE MAIN, CBSE, AIPMT Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App