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GO Classes DPP | GATE CS | Linear Algebra | Matrix Algebra

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Consider three non-zero matrices $A, ~B$ and $C$ such that $ABB^T = CBB^T$ where $B^T$ is the transpose of $B$. Which of the following statements is necessarily true?

  1. $r(A) =r(C)$

  2. non-zero eigenvalues of $A$ and $C$ are identical.

  3. $AB = CB$

  4. None of the above.

1 Answer

3 3 votes

Let $X=A-C$. Then the given condition becomes
$XBB^T=0$.

Now multiply on the right by $X'$:

$XBB^TX^T=0$.

But $XBB^TX^T=(XB)(XB)^T$.

So we have
$(XB)(XB)^T=0$.

A matrix of the form $YY^T$ is zero only when $Y=0$. Hence

$XB=0$.

That is,

$(A-C)B=0$,

so

$AB=CB$.

Therefore option C is necessarily true.

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