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

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  • 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

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