My question is about matrices.

Question

Hello, i have a question.

Suppose A is a 3*2 matrix, B is a p*q matrix and C is a 5*4 matrix. if ABC is defined then p and q is :

1) p=2 and q=5

2) p=2 and q=4

3) p=3 and q=4

4) p=3 and q=5

I originally went with answer number 1, but i am not sure if its correct or if my reasoning behind it is correct.

please share some insight to this question and its answer.

thank you

Answer 1

Yes, p = 2 and q = 5. To form the product AB, matrix B must have 2 rows (equalling the number of columns of A).

To form the product BC, matrix B must have 5 columns (equalling the number of rows of C). You can see this as follows: when you compute the i,j element of matrix BC you will multiply the first element of B's row i times the first element of C's column j. Then the second element of B's row i (which is in the second column of B) times the second element of C's column j (which is the second row of C). Separate the terms by a plus sign, of course, and so on. As you continue, you're grabbing one element from each column of B and you're grabbing one element from each row of C. If you run out of elements from one before the other, that wouldn't make sense. Therefore #columns of B must equal #rows of C.