Find the Characteristic Equation 2x-3i=[[7,4],[6,5]]

Problem

A=([7,4],[6,5])

Solution

1. Identify the matrix A for which the characteristic equation needs to be found.

A=([7,4],[6,5])

2. Recall the definition of the characteristic equation, which is given by the determinant of A−λ*I set to zero.

det(A−λ*I)=0

3. Subtract λ from the diagonal elements of matrix A to form the matrix A−λ*I

A−λ*I=([7−λ,4],[6,5−λ])

4. Calculate the determinant of the resulting 2×2 matrix by multiplying the diagonal elements and subtracting the product of the off-diagonal elements.

det(A−λ*I)=(7−λ)*(5−λ)−(4)*(6)

5. Expand the algebraic expression.

(7−λ)*(5−λ)−24=35−7*λ−5*λ+λ^2−24

6. Simplify the expression by combining like terms to obtain the characteristic polynomial.

λ^2−12*λ+11

7. Set the polynomial equal to zero to form the characteristic equation.

λ^2−12*λ+11=0

Final Answer

det(A−λ*I)=λ^2−12*λ+11=0

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