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A matrix $X$ has $(a+b)$ rows and $(a+2)$ columns; and a matrix $Y$ has $(b+1)$ rows and $(a+3)$ columns. If both $X Y$ and $Y X$ exist, then what are the values of $a, b$ respectively?
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The correct answer is:
2,3
The order of a given matrices are
$[X]_{(a+b)\times(a+2)}$ and $[Y]_{(b+1)\times(a+3)}$
As $[X Y]$ and $[Y X]$ exist
$\therefore a+2=b+1$ and $a+3=a+b$
$\Rightarrow a+3=a+b$
$\Rightarrow b=3$
Hence, $a=3+1-2=2$
$[X]_{(a+b)\times(a+2)}$ and $[Y]_{(b+1)\times(a+3)}$
As $[X Y]$ and $[Y X]$ exist
$\therefore a+2=b+1$ and $a+3=a+b$
$\Rightarrow a+3=a+b$
$\Rightarrow b=3$
Hence, $a=3+1-2=2$
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