Matrix multiplication: $X_{r \times c}$ and $Y_{c \times d}$

Question

Matrix $X$ has $r$ rows and $c$ columns, and matrix $Y$ has $c$ rows and $d$ columns, where $r, c$, and $d$ are different. Which of the following must be false?

1. The product $YX$ exists
2. The product of $XY$ exists and has $r$ rows and $d$ columns
3. The product $XY$ exists and has $c$ rows and $c$ columns

The answer says only 2 is false, but isn't 2 the only correct choice?

Answer 1

It helps to visualize matrix multiplication:

(courtesy of Wikimedia)

The number of columns of the first multiplicand has to match the number of rows of the second multiplicand.

Looking at the three choices:

1. The product YX exists
   This would require that $d$ the number of columns in Y equals the number $r$ of rows in X.
   Since $r \ne d$, this is not the case.
2. The product of XY exists and has $r$ rows and $d$ columns
   Since $c = c$, the product exists. It has in fact $r$ rows and $d$ columns.
3. The product XY exists and has $c$ rows and $c$ columns
   Since $c = c$, the product exists. But it has $r$ rows and $d$ columns.

Therefore, choice 2 is the only true one.