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 $A$.

 
LoadingAdd to solve later

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
\[\frac{\tr(A) \pm \sqrt{\tr(A)^2-4\det(A)}}{2}.\]


LoadingAdd to solve later

More from my site

You may also like...

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

More in Linear Algebra
Diagonalization Problems and Solutions in Linear Algebra
Find Eigenvalues, Eigenvectors, and Diagonalize the 2 by 2 Matrix

Consider the matrix $A=\begin{bmatrix} a & -b\\ b& a \end{bmatrix}$, where $a$ and $b$ are real numbers and $b\neq 0$....

Close