Math Explainer

Finding Eigenvalues with det(A − λI) = 0

Why must a determinant equal zero to find eigenvalues?

Lesson notes

Key points

Rewrite Av = λv as (A − λI)v = 0, where I is the identity matrix.
A homogeneous system has a nonzero solution only when its matrix is singular, meaning its determinant is zero.
For A with rows 2,1 and 1,2, det(A − λI) gives λ² − 4λ + 3 = 0.
Then substitute each eigenvalue back into (A − λI)v = 0 and solve for its eigenvectors.

Takeaway: Eigenvalues are the roots of det(A − λI) = 0.

Quick check: Why do we set det(A − λI) = 0?

To make A invertible
To allow nonzero solutions v
To make λ equal to zero

Only a singular A − λI lets (A − λI)v = 0 have a nonzero solution.

RearrangeSingular matrixCharacteristic polynomialBack-substitute

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