The Cayley-Hamilton Theorem

The Cayley-Hamilton Theorem is a remarkable result that states every square matrix satisfies its own characteristic equation. It has powerful applications for finding matrix inverses, computing matrix powers, and expressing higher powers of a matrix in terms of lower ones.

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Statement of the Theorem

Let $A$A be an $n \times n$n×n matrix with characteristic polynomial:

$p(\lambda) = \det(A - \lambda I)$p(λ)=det(A−λI)

The Cayley-Hamilton Theorem states:

$p(A) = O$p(A)=O

That is, when you substitute the matrix $A$A into its own characteristic polynomial (replacing $\lambda$λ with $A$A and the constant term with that constant times $I$I), the result is the zero matrix.

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2×2 Case

For a 2×2 matrix $A$A, the characteristic equation is:

$\lambda^2 - \text{tr}(A)\lambda + \det(A) = 0$λ2−tr(A)λ+det(A)=0

The Cayley-Hamilton Theorem gives:

$A^2 - \text{tr}(A)\cdot A + \det(A)\cdot I = O$A2−tr(A)⋅A+det(A)⋅I=O

Worked Example 1: Verify the Cayley-Hamilton Theorem for $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$A=(13​24​).

$\text{tr}(A) = 5$tr(A)=5, $\det(A) = 4 - 6 = -2$det(A)=4−6=−2.

Characteristic equation: $\lambda^2 - 5\lambda - 2 = 0$λ2−5λ−2=0.

Cayley-Hamilton says: $A^2 - 5A - 2I = O$A2−5A−2I=O.