## Precalculus (6th Edition) Blitzer

a. $a^2$ b. $a^3$ c. $a^4$ d. see explanations. e. see explanations.
a. $\begin{vmatrix} a & a \\0 & a \end{vmatrix}=a(a)-a(0)=a^2$ b. Using expansion from the first column, we have: $\begin{vmatrix} a & a & a \\0 & a & a \\0 & 0 & a \end{vmatrix}=(a)\begin{vmatrix} a & a \\0 & a \end{vmatrix}-0+0=(a)(a^2)=a^3$ c. Using expansion from the first column, we have: $\begin{vmatrix} a & a & a & a \\0 & a & a & a \\0 & 0 & a & a\\0 & 0 & 0 & a \end{vmatrix}=(a)\begin{vmatrix} a & a & a \\0 & a & a \\0 & 0 & a \end{vmatrix}-0+0-0=(a)(a^3)=a^4$ d. The determinant contains only $a$ and $0$. Draw a line diagonally (top left to bottom right). All the elements below this line are zeros. e. The result is $a^n$, where $n$ is the number of elements in a row.