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The matrix A has $x$ rows and $x+5$ columns. The matrix $B$ has y rows and $11-y$ columns. Both $\mathrm{AB}$ and BA exist. What are the values of $\mathrm{x}$ and $\mathrm{y}$ respectively?
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The correct answer is:
3 and 8
Given, Matrix A has $x$ rows and $x+5$ columns Matrix $\mathrm{B}$ has $\mathrm{y}$ rows and $11-\mathrm{y}$ columns. Also given $\mathrm{AB}$ and $\mathrm{BA}$ exists. If AB exists, then the number of rows in A most be equal to number of columns in $\mathrm{B}$.
i.e., $x=11-y$ $\ldots .(1)$
If BA exists, then the number of rows in $\mathrm{B}$ must be equal to number of rows in $\mathrm{A}$.
i.e., $x+5=y$ $\Rightarrow 11-y+5=y($ from $(1))$
$\Rightarrow 2 y=16$
$\Rightarrow y=8$
$(1) \Rightarrow x=11-8=3$
So, $x=3, y=8$.
i.e., $x=11-y$ $\ldots .(1)$
If BA exists, then the number of rows in $\mathrm{B}$ must be equal to number of rows in $\mathrm{A}$.
i.e., $x+5=y$ $\Rightarrow 11-y+5=y($ from $(1))$
$\Rightarrow 2 y=16$
$\Rightarrow y=8$
$(1) \Rightarrow x=11-8=3$
So, $x=3, y=8$.
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