The eigenvalues of $A = \left( \begin{array} { c c }
2 & - 2 \
2 & 2
\end{array} \right)$ is
A) $0,4$
B) $0 , - 4$
C) $2,2$
D) $2 \pm 2 i$
E) $- 2 \pm 2 i$

Question

The eigenvalues of $A = \left( \begin{array} { c c } 
2 & - 2 \
2 & 2
\end{array} \right)$ is
A) $0,4$
B) $0 , - 4$
C) $2,2$
D) $2 \pm 2 i$
E) $- 2 \pm 2 i$

Quizplus · Accepted Answer

The eigenvalues of a matrix $A$ are found by solving the characteristic equation $\det(A - \lambda I) = 0$, where $I$ is the identity matrix and $\lambda$ represents the eigenvalues. For the given matrix $A$, the characteristic equation is $\det\left( \begin{array}{cc} 2-\lambda & -2 \ 2 & 2-\lambda \end{array} \right) = 0$, which simplifies to $(2-\lambda)^2 - (-2)(2) = \lambda^2 - 4\lambda + 4 - 4 = \lambda^2 - 4\lambda = 0$. Factoring this gives $\lambda^2 - 4\lambda + 4 = (\lambda - 2)^2 = 0$, indicating a mistake in my calculation. Let's correct this: the correct expansion should include the +4 from the original determinant calculation, leading to $\lambda^2 - 4\lambda + 4 = (\lambda - 2)^2 - 4$, which correctly simplifies to $\lambda^2 - 4\lambda + 4 = \lambda^2 - 4\lambda + 4 = (\lambda - 2)^2 = 0 + 4$. Correcting my mistake, the determinant calculation should actually yield $\lambda^2 - 4\lambda + 8 = 0$, solving this quadratic equation gives $\lambda = 2 \pm 2i$, which corresponds to option D. My initial explanation contained errors in the determinant calculation and solving the characteristic equation. The correct process involves finding the determinant of $A - \lambda I$, leading to a quadratic equation whose solutions are the eigenvalues of $A$.

The Eigenvalues Of A=(2−222)A = \left( \begin{array} { c c } 2 & - 2 \\2 & 2\end{array} \right)A=(22​−22​)

The Eigenvalues Of $A = \left( \begin{array} { c c } 2 & - 2 \\2 & 2\end{array} \right)$