# How does one multiply a column vector by a row vector?

I was trying to solve some previous year questions from a screening test for engineers in India, GATE and I came across this question where they tried to define a matrix as,

$$A=\begin{bmatrix}2 \\-4 \\ 7\end{bmatrix}\begin{bmatrix}1&9&5\end{bmatrix}$$

I have never seen a matrix being defined in this way. As far as I know such a multiplication is not possible. I am not an engineer and I am unaware of such notations that engineers use. So, what is the matrix that is being represented here?

• Think of the vectors as matrices. – Angina Seng Jun 30 '19 at 11:24
• 3 cross 3, perhaps. – XRFXLP Jun 30 '19 at 11:25
• "As far as I know such a multiplication is not possible.": Do you know the definition of matrix multiplication? If $A$ is $m\times k$ and $B$ is $k\times n$ then $AB$ is defined (and is $m\times n$). Here you have a $3\times 1$ matrix times a $1\times 3$ matrix - what's the problem? – David C. Ullrich Jun 30 '19 at 15:04

It is the matrix$$\begin{bmatrix}2\times1&2\times9&2\times5\\-4\times1&-4\times9&-4\times5\\7\times1&7\times9&7\times5\end{bmatrix}.$$This is the standard product of two matrices.