Question

verify Cayley Hamilton theorem
and hence find inverse of matrix {2 -1 1,-1 2 -1,1 -1 2}​

Answer 1

Check the attached file.

Matrices .

Answer 2

Given: matrix {2 -1 1 , -1 2 -1, 1, -1 2}

To Find : a) Cayley Hamilton theorem

               b) inverse of the given Matrix

Step-by-step explanation:

a) Cayley Hamilton theorem states that every square matrix over its commutative ring satisfies its own characteristic equation.

Let us look this through an example

• $A$= $\left[\begin{array}{ccc}1&2\\3&4\end{array}\right]$

• p(λ) =det(λI₂−A) =det (λ−1  −2                                  
•                                   −3     λ-4)

                                       

• =(λ−1)(λ−4) −(−2)(−3) = λ²−5λ−2

• The Cayley-Hamilton claims that if, we define
• p(X) = X²- 5X-2I₂
• then,
• p(A) = A²-5A-2I₂=$\left[\begin{array}{ccc}0&0\\0&0\end{array}\right]$

• We can verify this result by computation
• A²-5A-2I₂ =$\left[\begin{array}{ccc}7&10\\15&22\end{array}\right]$  -$\left[\begin{array}{ccc}5&10\\15&20\end{array}\right]$ -$\left[\begin{array}{ccc}2&0\\0&2\end{array}\right]$=$\left[\begin{array}{ccc}0&0\\0&0\end{array}\right]$
• For a generic 2 x 2 matrix,
• A= $\left[\begin{array}{ccc}a&b\\c&d\end{array}\right]$
• the resultant polynomial is given by
• P(λ) = λ²−(a+d)λ+(ad−bc) , so the Cayley-Hamilton theorem states that
• p(A) = A²−(a + d )A+(ad−bc)I₂=$\left[\begin{array}{ccc}0&0\\0&0\end{array}\right]$
• it is always the case, which is evident by working out on

      A².

Inverse of the Given Matrix

{3/4  1/4  -1/4, 1/4 3/4 1/4, -1/4 1/4 3/4}