Cayley-Hamilton Theorem

Rarely Tested

Cayley-Hamilton Theorem

## Definition / Concept

Every square matrix satisfies its own characteristic equation.

For a $$2\times2$$ matrix:

$$A^2-(\operatorname{tr}A)A+|A|I=O$$

where $$\operatorname{tr}A$$ is the trace of $$a$$.

## Usage

- Used to reduce higher powers of a matrix to lower powers.

Question 1

Consider the matrix $$M=\begin{bmatrix}2&-1\\1&0\end{bmatrix}.$$

Let $$p,q,r,s,a,b,c$$ and $$d$$ be integers such that $$M^{26}=\begin{bmatrix}p&q\\r&s\end{bmatrix}$$ and $$\displaystyle\sum_{k=1}^{26}M^k=\begin{bmatrix}a&b\\c&d\end{bmatrix}.$$

Then which of the following statements is (are) TRUE?

Question 2

Let $$A= \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} $$ and $$I$$ be the identity matrix of order 2. If $$A^2-5A+kI=O$$ then the value of $$k$$ is :

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