Which of the following statements is (are) true, for three matrices?

Question

Let $A, B, C$ be three matrices such that $AB = C$. Which of the following statements is (are) true?

1. the columns in $C^T$ are linear combinations of the columns in $B^T$

2. the columns in $C$ are linear combinations of the columns in $A^T$

3. the columns in $C$ are linear combinations of the columns in $B$

4. the columns in $C^T$ are linear combinations of the columns in $A^T$

My answer: I guess that the only option that is correct is 3 which means that The columns in C are linear combinations of the columns in B. Am I right or could someone please help me to decide of which of these that are correct?

Answer 1

Hint: The matrix $AB$ has columns that are the linear combinations of the columns of $A$, and rows that are the linear combinations of the rows of $B$. To see that this is true, note that each column of $AB$ has the form $$ \pmatrix{C_1 & C_2 & C_3} \pmatrix{b_1\\b_2\\b_3} = b_1 C_1 + b_2 C_2 + b_3 C_3 $$ where $C_i$ is the $i$th column of $A$. Similarly, each row of $AB$ as the form $$ \pmatrix{a_1 & a_2 & a_3} \pmatrix{R_1\\R_2\\R_3} = a_1 R_1 + a_2 R_2 + a_3 R_3 $$ where $R_i$ is the $i$th row of $B$.

Of course, the columns of $A^T$ are simply the rows of $A$.