Express the Eigenvalues of a 2 by 2 Matrix in Terms of the Trace and Determinant

Linear algebra problems and solutions

Problem 631

Let $A=\begin{bmatrix}
a & b\\
c& d
\end{bmatrix}$ be an $2\times 2$ matrix.

Express the eigenvalues of $A$ in terms of the trace and the determinant of $A$.

 
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Solution.

Recall the definitions of the trace and determinant of $A$:
\[\tr(A)=a+d \text{ and } \det(A)=ad-bc.\]


The eigenvalues of $A$ are roots of the characteristic polynomial $p(t)$ of $A$. So let us first find $p(t)$.
We have
\begin{align*}
p(t) &= \det(A-tI)=\begin{vmatrix}
a-t & b\\
c& d-t
\end{vmatrix}\\[6pt] &=(a-t)(d-t)-bc\\
&=t^2-(a+d)t+ad-bc\\
&=t^2-\tr(A) t+\det(A).
\end{align*}

Using the quadratic formula, the eigenvalues of A (roots of $p(t)$) are
\[\frac{\tr(A) \pm \sqrt{\tr(A)^2-4\det(A)}}{2}.\]


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