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?

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.