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.