Let p and q be real numbers such that p² + q² = 1. The eigenvalues of the matrix $\left[\begin{array}{cc}p& q\\ q& -p\end{array}\right]$ are

The correct answer is 1 and -1.

To find the eigenvalues of the given matrix, we can set up and solve its characteristic equation. Let the matrix be denoted by A:

$A=\left[\begin{array}{cc}p& q\\ q& -p\end{array}\right]$

The characteristic equation is given by det(A - λI) = 0, where λ represents the eigenvalues and I is the identity matrix of order 2. This can be written as:

$\left|\begin{array}{cc}p-\lambda & q\\ q& -p-\lambda \end{array}\right|=0$

Expanding the determinant, we get:
$(p-\lambda )(-p-\lambda )-\left(q\right)\left(q\right)=0$

Simplifying the expression:
$-(p-\lambda )(p+\lambda )-q^2=0$
$-(p^2-{\lambda }^2)-q^2=0$
${\lambda }^2-(p^2+q^2)=0$

We are given that p² + q² = 1. Substituting this value into the equation:
${\lambda }^2-1=0$
${\lambda }^2=1$
$\lambda =\pm 1$

Therefore, the eigenvalues of the matrix are 1 and -1.