# Simplifying expressions

Is it mathematically valid to simplify the expression $$\left ( \bar \Psi \right)^2 \left(1, \ -1\right)$$ to 1 if $$\Psi = \begin{pmatrix} \cos(x) & 0 \\ 0 & i \sin(x) \end{pmatrix}$$ (where $$\bar \Psi$$ is the complex conjugate of $$\Psi$$)? Since \begin{align} \left( \bar \Psi \right)^2 \left(1,\ -1\right) &= \bar \Psi \bar \Psi \left(1,\ -1\right) &= \bar \Psi \Psi \\ &= 1 \end{align}

• Use mathjax. Please do not post images of math. – DanielSank Nov 19 '19 at 9:01

You have $$\bar\Psi = \begin{pmatrix} \cos(x) & 0 \\ 0 & -i \sin(x) \end{pmatrix}.$$ Then, $$\bar\Psi\bar\Psi= \begin{pmatrix} \cos^2(x) & 0 \\ 0 & -\sin^2(x) \end{pmatrix}.$$ This yields $$\bar\Psi\bar\Psi\begin{pmatrix}1 \\ -1\end{pmatrix}= \begin{pmatrix} \cos^2(x) & 0 \\ 0 & -\sin^2(x) \end{pmatrix}\begin{pmatrix}1 \\ -1\end{pmatrix}= \begin{pmatrix}\cos^2(x) \\ \sin^2(x)\end{pmatrix}.$$