How Do You Find the Characteristic Polynomial of a 2×2 Matrix?

For a $2\times 2$ matrix $A$, calculate the characteristic polynomial using $p(\lambda)=\det(A-\lambda I)$. If $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, then $p(\lambda)=\lambda^2-(a+d)\lambda+(ad-bc)$, and the roots of $p(\lambda)=0$ are the eigenvalues.

The Reasoning

Start with the eigenvector equation $A\mathbf{v}=\lambda\mathbf{v}$. Rearranging gives

$$(A-\lambda I)\mathbf{v}=\mathbf{0}.$$

An eigenvector must be non-zero, so this homogeneous system needs a non-trivial solution. Therefore, $A-\lambda I$ must be singular, meaning its determinant is zero.

Subtract $\lambda$ from the diagonal entries only:

$$A-\lambda I=\begin{pmatrix}a-\lambda&b\\c&d-\lambda\end{pmatrix}.$$

Expanding the determinant gives

$$\begin{aligned} p(\lambda)&=(a-\lambda)(d-\lambda)-bc\\ &=\lambda^2-(a+d)\lambda+(ad-bc). \end{aligned}$$

This provides two useful checks. The coefficient of $\lambda$ is the negative trace, where $\operatorname{tr}(A)=a+d$, while the constant term is $\det(A)=ad-bc$. Consequently, the eigenvalues have sum and product .

Worked Example

Let

$$A=\begin{pmatrix}4&2\\1&3\end{pmatrix}.$$

Then

$$\begin{aligned} p(\lambda)&=\det\begin{pmatrix}4-\lambda&2\\1&3-\lambda\end{pmatrix}\\ &=(4-\lambda)(3-\lambda)-2\\ &=\lambda^2-7\lambda+10. \end{aligned}$$

Set the characteristic polynomial equal to zero:

$$\lambda^2-7\lambda+10=(\lambda-5)(\lambda-2)=0.$$

Hence, the eigenvalues are $5$ and $2$. Their sum is $7$, matching the trace, and their product is $10$, matching the determinant.

A common misconception is to subtract $\lambda$ from every entry. This is incorrect because $\lambda I=\begin{pmatrix}\lambda&0\\0&\lambda\end{pmatrix}$, so only diagonal entries change. Another frequent error is writing instead of the correct determinant (ad-bc).

IB Exam Technique

This is HL-only content from AHL 1.15, Eigenvalues and eigenvectors. Show $A-\lambda I$, the determinant equation, and the expanded polynomial before solving. Use your GDC to check roots when appropriate, but present the characteristic equation in mathematical notation and retain exact values where possible.