Question

Let A be a 3 × 4 matrix, B be a 4 × 5 matrix, and C be a 4 × 4 matrix. Determine which of the following productsare defined and find the size of those that are defined.

a) AB b) BA c) AC d) CA e) BC f ) CB

Short answer

Answers for each part are given below:

a) 3 x 5

b) Not define

c) 3 x 4

d) Not define

e) Not define

f) 4 x 5

Step-by-step solution

Step1:

Multiplication of two matrices P and Q i.e., PQ is defined when number of columns of P is equal to the number of rows of Q.

Step2:

1. The matrix product AB is defined as the inner dimension of the two matrices is both 4 and AB is a 3 x 5 matrix.
2. The matrix product BA is not defined as the inner dimension of the two matrices are not equal, since $5\ne 3$
3. The matrix product AC is defined as the inner dimension of the two matrices is both 4 and AC is then a 3 x 4 matrix.
4. The matrix product CA is not defined as the inner dimension of the two matrices are not equal since $4\ne 3$.
5. The matrix product BC is not defined as the inner dimension of the two matrices are not equal, since$5\ne 4$
6. The matrix product CB is defined as the inner dimension of the two matrices are equal and CB is the5n a 4 x 5 matrix.