# Matrix multiplication - Express a column as a linear combination

Let $A = \begin{bmatrix} 3 & -2 & 7\\ 6 & 5 & 4\\ 0 & 4 & 9 \end{bmatrix}$ and $B = \begin{bmatrix} 6 & -2 & 4\\ 0 & 1 & 3\\ 7 & 7 & 5 \end{bmatrix}$

Express the third column matrix of $AB$ as a linear combination of the column matrices of $A$

I don't get this... surely the 3rd column would be an expression of the row matrices of $A$ since the 3rd column of$AB$ would be $\begin{bmatrix} 3(4) & -2(3) & 7(5)\\ 6(4) & 5(3) & 4(5)\\ 0(4) & 4(3) & 9(5) \end{bmatrix}$

As I typed out the question I see my answer...

the 3rd column is $4\begin{bmatrix} 3\\ 6\\ 0 \end{bmatrix} 3 \begin{bmatrix} -2\\ 5\\ 4 \end{bmatrix} 5 \begin{bmatrix} 7\\ 4\\ 9 \end{bmatrix}$

Is this correct?

• The $3$rd column of $AB$, as you've written it, looks like a $3\times 3$ matrix, not a $3\times 1$ matrix (i.e. a column vector). Oct 21, 2013 at 19:40

Not quite: we need to add entries. So the third column of matrix $AB$ is given by:
So the third column represented as a linear combination of columns of $A$ is given by:
$$4 \begin{bmatrix} 3\\ 6\\ 0 \end{bmatrix} + 3 \begin{bmatrix} -2\\ 5\\ 4 \end{bmatrix} +5 \begin{bmatrix} 7\\ 4\\ 9 \end{bmatrix}$$
The third column of $AB$ is $$A\begin{bmatrix} 4\\ 3\\ 5 \end{bmatrix}$$ and if we denote the columns of $A=[C_1 C_2 C_3]$ then the third column of $AB$ is $$A\begin{bmatrix} 4\\ 3\\ 5 \end{bmatrix}=[C_1 C_2 C_3]\begin{bmatrix} 4\\ 3\\ 5 \end{bmatrix}=4C_1+3C_2+5C_3$$