Row × Column Matrix Products

Solution:

Part 1: \( [a\;\; b]\begin{bmatrix}c\\ d\end{bmatrix} \)

This is a dot product:

\[ = ac + bd \]

Part 2: \( [a\;\; b\;\; c\;\; d]\begin{bmatrix}a\\ b\\ c\\ d\end{bmatrix} \)

\[ = a^2 + b^2 + c^2 + d^2 \]

Final Answer:

\[ [a\;\; b]\begin{bmatrix}c\\ d\end{bmatrix} = ac + bd \] \[ [a\;\; b\;\; c\;\; d]\begin{bmatrix}a\\ b\\ c\\ d\end{bmatrix} = a^2 + b^2 + c^2 + d^2 \]