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Let A and B be two matrices.Which of the following statements is TRUE ?

       S1 : If product of A and B ( (i.e) AB ) = 0-matrix ,then either one of A or B should be equal to 0-matrix.

       S2 : If product of A and B ( (i.e) AB ) = A , then B should be an identity matrix.

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Best answer
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$S1:-$ The equation $AB=0$ doesn't necessarily imply that one of the matrices A and B must be Zero.

Example :-  $A= \begin{bmatrix} 1 & 1\\ -1 & -1 \end{bmatrix}$

$B= \begin{bmatrix} -1 & -1\\ 1 & 1 \end{bmatrix}$

$AB= \begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix}$

Here both A and B are not $Zero$ matrices but Multiplication of $A$ and $B$ is $ZERO$.

Hence,$S1$ is False Statement.

$S2:-$ If product of A and B ( (i.e) AB ) = A , then B should be an identity matrix.

Example:- $A=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$

$B=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

$AB=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}=A$

Hence,Statement $S2$ is True.

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S1 may not be true , as either of the matrix can have negative numbers , which may give result as 0.

 

S2 should be true.

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