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.

Here,

$[X]_{(a+b) \times(a+2)}$

$[Y]_{(b+1) \times(a+3)}$

Since $X Y$ exists, the number of columns in $X$ is equal to the number of rows in $Y$.

$\Rightarrow a+2=b+1 \quad \ldots(1)$

Similarly, $\sin c_{e} Y X$ 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.