Answer
a. $a^2$
b. $a^3$
c. $a^4$
d. see explanations.
e. see explanations.
Work Step by Step
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.
