Question

If A = [a b] and B = `[(c), (d)]`, then:

Options

• only matrix AB is possible.

• only matrix BA is possible.

• both matrices AB and BA are possible.

• both matrices AB and BA are possible, AB = BA.

Answer 1

Both matrices AB and BA are possible.

Explanation:

A = [a b] has order 1 × 2 and B = `[(c), (d)]` has order 2 × 1.

Since the number of columns of A(2) equals the number of rows of B(2), AB is defined and has order 1 × 1: AB = ac + bd.

Also B(2 × 1) times A(1 × 2) is defined and gives a 2 × 2 matrix BA = `[[ac, bc],[ad, bd]]`. 

Because AB is a 1 × 1 scalar and BA is a 2 × 2 matrix, they are not equal in general.

This follows from the usual rule for matrix multiplication product defined when cols (A) = rows (B).