# Find the Eigenvalues [[0.1,0.9],[0.4,0.6]] Set up the formula to find the characteristic equation .
Substitute the known values in the formula.
Subtract the eigenvalue times the identity matrix from the original matrix.
Multiply by each element of the matrix.
Simplify each element of the matrix .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Add the corresponding elements of to each element of .
Simplify each element of the matrix .
Simplify .
Simplify .
The determinant of is .
These are both valid notations for the determinant of a matrix.
The determinant of a matrix can be found using the formula .
Simplify the determinant.
Simplify each term.
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Simplify each term.
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Subtract from .
Multiply by .
Subtract from .
Set the characteristic polynomial equal to to find the eigenvalues .
Solve the equation for .
Use the quadratic formula to find the solutions.
Substitute the values , , and into the quadratic formula and solve for .
Simplify.
Simplify the numerator.
Raise to the power of .
Multiply by .
Multiply by .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Multiply by .
Simplify .
Simplify the expression to solve for the portion of the .
Simplify the numerator.
Raise to the power of .
Multiply by .
Multiply by .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Multiply by .
Simplify .
Change the to .
Simplify the expression to solve for the portion of the .
Simplify the numerator.
Raise to the power of .
Multiply by .
Multiply by .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Multiply by .
Simplify .
Change the to .
Subtract from .
The final answer is the combination of both solutions.
Find the Eigenvalues [[0.1,0.9],[0.4,0.6]]

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