# A matrix X

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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