### Instruction: The matrix
\[\begin{pmatrix} a & b \\ -\frac{4}{5} & \frac{3}{5} \end{pmatrix}\]corresponds to a reflection.  Enter the ordered pair $(a,b).$.### Response: A matrix corresponds to a reflection if its columns are orthonormal, that is, they have unit length and are orthogonal to each other.
So, I need to find $a$ and $b$ such that 
\[\begin{pmatrix} a \\ -\frac{4}{5} \end{pmatrix}\cdot \begin{pmatrix} b \\ \frac{3}{5} \end{pmatrix}=0\]and
\[\begin{pmatrix} a \\ -\frac{4}{5} \end{pmatrix}\cdot \begin{pmatrix} a \\ -\frac{4}{5} \end{pmatrix}=1\]and
\[\begin{pmatrix} b \\ \frac{3}{5} \end{pmatrix}\cdot \begin{pmatrix} b \\ \frac{3}{5} \end{pmatrix}=1.\]
The first equation gives me $ab-\frac{12}{25}=0$, or $ab=\frac{12}{25}$.
The second equation gives me $a^2+\frac{16}{25}=1$, or $a^2=\frac{9}{25}$.
The third equation gives me $b^2+\frac{9}{25}=1$, or $b^2=\frac{16}{25}$.
Taking the positive square roots of the last two equations, I get $a=\frac{3}{5}$ and $b=\frac{4}{5}$.
(I could also take the negative roots, but then the matrix would correspond to a different reflection.)