Question

Let A be a 3 by 6 , B be a 6 by 7 and C be a 7 by 3 matrix. Determine the size of the following matrices (if they do not exist, type N in both answer boxes):

a. AB: ______ by ________
b. BA: ______ by ________
c. A^TB: _______ by ________
d. BC: __________ by ________

Answer 1

a. AB: 3 by 7

b. BA: N by N

c. A^TB: N by N

d. BC: 6 by 3

Explanation:

Given

`A =3\ by\ 6`

`B =6\ by\ 7`

`C =7\ by\ 3`

Required

The dimension of the following matrices

As a general rule:

For A * B to be successful, the columns in a must equal the rows in B

Using this rule, we have:

`A_(m*n) * B_(n * p) = AB_(m*p)`

So:

`(a)\ AB`

`A_(3*6) * B_(6*7) \to AB_(3 * 7)`

`(b)\ BA`

`B_(6*7) * A_(3*6) \to AB_(N * N)`

The column numbers of B does not equal the row numbers of A.

Hence, BA does not exist

`(c)\ A^TB`

`A^T` implies that:

If
`A =3\ by\ 6`, then

`A^T = 6\ by\ 3`

So:

`A^T_(6*3) * B_(6,7) \to A^TB_(N*N)`

The column numbers of A^T does not equal the row numbers of B.

Hence,
`A^TB` does not exist

`(d)\ BC`

`B_(6*7) * C_(7*3) \to BC_(6,3)`