Find the Characteristic Equation (A^t-6I)^-1=[[0,1],[1,0]]

Problem

(AT−6*I)(−1)=[[0,1],[1,0]]

Solution

1. Invert both sides to solve for the expression inside the inverse. Since the inverse of [[0,1],[1,0]] is itself, we have:

AT−6*I=[[0,1],[1,0]]

2. Isolate AT by adding 6*I to both sides, where I=[[1,0],[0,1]]

AT=[[0,1],[1,0]]+[[6,0],[0,6]]

AT=[[6,1],[1,6]]

3. Find matrix A by taking the transpose of AT Since the matrix is symmetric:

A=(AT)T=[[6,1],[1,6]]

4. Set up the characteristic equation using the formula det(A−λ*I)=0

det(6−λ)=0

5. Calculate the determinant by multiplying the diagonal elements and subtracting the product of the off-diagonal elements:

(6−λ)*(6−λ)−(1)*(1)=0

6. Expand and simplify the quadratic expression:

36−12*λ+λ2−1=0

λ2−12*λ+35=0

Final Answer

λ2−12*λ+35=0

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