Answer :

X has a + b rows and a + 2 columns.


Order of X = (a + b) × (a + 2)


Y has b + 1 rows and a + 3 columns.


Order of Y = (b + 1) × (a + 3)


Recall that the product of two matrices A and B is defined only when the number of columns of A is equal to the number of rows of B.


It is given that the matrix XY exists.


Number of columns of X = Number of rows of Y


a + 2 = b + 1


a = b – 1


The matrix YX also exists.


Number of columns of Y = Number of rows of X


a + 3 = a + b


b = 3


We have a = b – 1


a = 3 – 1


a = 2


Thus, a = 2 and b = 3.


Hence, order of X = 5 × 4 and order of Y = 4 × 5.


Order of XY = Number of rows of X × Number of columns of Y


Order of XY = 5 × 5


Order of YX = Number of rows of Y × Number of columns of X


Order of XY = 4 × 4


As the orders of the two matrices XY and YX are different, they are not of the same type and thus unequal.


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