Question

If Matrix A has dimensions 1x4 and Matrix B has dimensions 3x4, can these be multiplied?

Answer 1

Answer: No. You cannot multiply the matrices in any order. A*B is not defined. Also, B*A is not defined.

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Explanation: 

Matrix A has 1 row, and 4 columns. 
Matrix B has 3 rows, and 4 columns

In order for A*B to be possible, A has to have the same number as columns as B has rows. In other words, the inner dimensions have to match up. The '4' in '1x4' needs to match up with the '3' in the '3x4'. This match doesn't happen. 

The same story happens with B*A, just things have been flipped. 

B has 3 rows, 4 columns
A has 1 row, 4 columns

The "4 columns" of B does not match with "1 row" of A. 

Answer 2

$\bf A= \begin{bmatrix} \square &\square &\square &\square \end{bmatrix}\qquad \qquad B= \begin{bmatrix} \square &\square &\square &\square \\ \square &\square &\square &\square \\ \square &\square &\square &\square \\ \end{bmatrix}$

notice above, the matrix A has 1 row 4 columns, a 1x4,
and the B matrix has 3 rows and 4 columns, 3x4.

since B has only 3 rows, not 4, no dice.