Medium Practice — Set 1

Quick Recap — Matrix Algebra

• Multiplication is row $\times$ column; not commutative in general; associative.
• $A+A^T$ is symmetric; $A-A^T$ is skew-symmetric.
• Inverse ($2\times2$): $A^{-1}=\tfrac{1}{|A|}\operatorname{adj}A$, with $\operatorname{adj}\begin{pmatrix}a&b\\c&d\end{pmatrix}=\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$.
• Powers of $\begin{pmatrix}1&1\\0&1\end{pmatrix}$: the $(1,2)$-entry counts the exponent.