Answer
$a.\qquad A^{2}=\left[\begin{array}{ll}
4 & -45\\
0 & 49
\end{array}\right]$
$ b.\qquad A^{3}=\left[\begin{array}{ll}
8 & -335\\
0 & 343
\end{array}\right]$
Work Step by Step
If $A$ is an $m\times n$ matrix and $B$ is an $n\times k$ matrix
(so the number of columns of $A$ is the same as the number of rows of $B$),
then the matrix product $AB$ is the $m\times k$ matrix
whose $ij$-entry is the inner product of the $i\mathrm{t}\mathrm{h}$ row of $A$ and the jth column of $B.$
---
$a.$
$A$ is a 2$\times$2 matrix, so $AA=A^{2}$ is defined, and is 2$\times$2 matrix.
$A^{2}=\left[\begin{array}{ll}
2(2)+(-5)(0) & 2(-5)+(-5)(7)\\
0(2)+7(0) & 0(-5)+7(7)
\end{array}\right]=\left[\begin{array}{ll}
4 & -45\\
0 & 49
\end{array}\right]$
$b.$
$A^{3}=AA^{2}$ is defined, and is 2$\times$2 matrix.
$A^{3}=\left[\begin{array}{ll}
2(4)+(-5)(0) & 2(-45)+(-5)(49)\\
0(4)+7(0) & 0(-45)+7(49)
\end{array}\right]=\left[\begin{array}{ll}
8 & -335\\
0 & 343
\end{array}\right]$
