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

Answer:

a. AB: 3 by 7

b. BA: N by N

c. A^TB: N by N

d. BC: 6 by 3

Step-by-step 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

$(d)\ BC$

$B_{6*7} * C_{7*3} \to BC_{6,3}$