A matrix X has a + b rows and a + 2 columns while the matrix Y ha… - Vidyadip

Question

A matrix $X$ has $a + b$ rows and $a + 2$ columns while the matrix $Y$ has $b + 1$ rows and $a + 3$ columns. Both matrices $XY$ and $YX$ exist. Find $a$ and $b.$ Can you say $XY$ and $YX$ are of the same type $?$ Are they equal.

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Answer

Here, $[X]_{(a+b) \times (a+2)} [Y]_{(b+1) \times (a+3)}$
Since $XY$ exists, the number of columns in $X$ is equal to the number of rows in $Y. $
$\Rightarrow a + 2 = b + 1 ...(1)$
Similarly, since $YX$ exists, the number of columns in $Y$ is equal to the number of rows in $X.$
$\Rightarrow a + b = a + 3 $
$\Rightarrow b = 3$ Putting the value of $b$ in $(1),$
we get $a + 2 ≈ 3 + 1 $
$\Rightarrow a = 2$
Since the order of the matrices $XY$ and $YX$ is not same,
$XY$ and $YX$ are not of the same type and they are unequal.

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