Skill: Multiply a row matrix by a column matrix

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Multiplying a $1\times n$ row matrix by an $n\times1$ column matrix pairs entries in corresponding positions, multiplies each pair, and adds the products; the compatible dimensions produce a $1\times1$ matrix, identified with a scalar. This operation is the dot product of the two ordered lists, not entrywise multiplication; the treatment is limited to finite matrices with real entries, not more general algebraic settings.

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Detailed Explanation: Multiply a row matrix by a column matrix

To multiply a $1\times n$ row matrix by an $n\times 1$ column matrix, multiply the entries in matching positions and add the products. The result is a $1\times 1$ matrix, which we can write as a scalar.

For example, multiply

\begin{bmatrix}2 & -1 & 4\end{bmatrix} \begin{bmatrix}3\\5\\-2\end{bmatrix}.

Pair the entries in order, multiply each pair, then add:

(2)(3)+(-1)(5)+(4)(-2) =6-5-8 =-7.

So the product is the $1\times 1$ matrix $\begin{bmatrix}-7\end{bmatrix}$ , or simply the scalar $-7$ . This is a dot product—not entrywise multiplication.

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