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The question is :

If a square matrix $A$ is row-equivalent to a matrix $B$ but not congruent to $B$ then can it be said that two matrices have the same minimal polynomial?Please help me.

Thank you in advance.

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No. The row operations preserve the determinant, but almost none else.

For example, take the matrices $$ I = \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \qquad J = \begin{pmatrix} 1 & 1\\ 0 & 1 \end{pmatrix}. $$

They are row equivalent, but their minimal polynomials are different $p_I(x)=1-x$, $p_J(x) = (1-x)^2$.

Call $A$ the original matrix. Let's build a row-equivalent but not congruent matrix $B$.

Modify $A$ into an upper triangular form $T$ with at least one non-zero element on the diagonal (you can always do so, except when $A$ is the zero matrix). Then multiply all the rows by a gigantic number $a$, so that the determinant of $aT$ is different from the determinant of $A$. Then $B=aT$.

If you don't want to use the rows rescaling or swapping(that modify the determinant) is a bit more complicated..

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    I also have a problem though it is not fully related to it.The question is what is the easiest way to construct a matrix $A$ which is row equivalent to a matrix $B$ but not congruent to $B$.I think my question is too silly.But I want to clear my concept. – 2017-02-20
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    Call $A$ the original matrix. Bring it to an upper triangular form $T$ with at least one non-zero element on the diagonal (you can always do so, except when $A$ is the zero matrix). Then multiply all the rows by a gigantic number (for example the double of the sum of all the absolute values of entries in $A$). – 2017-02-20
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    Notice that the zero matrix can't be transformed through row operations into something else. – 2017-02-20