# Find the Eigenvalues [[4,0,1],[2,3,2],[49,0,4]]

Set up the formula to find the characteristic equation .

Substitute the known values in the formula.

Multiply by each element of the matrix.

Simplify each element of the matrix .

Rearrange .

Rearrange .

Rearrange .

Rearrange .

Rearrange .

Rearrange .

Rearrange .

Rearrange .

Rearrange .

Add the corresponding elements of to each element of .

Simplify each element of the matrix .

Simplify .

Simplify .

Simplify .

Simplify .

Simplify .

Simplify .

Set up the determinant by breaking it into smaller components.

Since the matrix is multiplied by , the determinant is .

The determinant of is .

The determinant of a matrix can be found using the formula .

Simplify the determinant.

Simplify terms.

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 .

Expand by multiplying each term in the first expression by each term in the second expression.

Simplify terms.

Simplify each term.

Multiply by by adding the exponents.

Move .

Multiply by .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Multiply by .

Multiply by .

Multiply by .

Multiply by .

Multiply by .

Simplify by adding terms.

Add and .

Subtract from .

Since the matrix is multiplied by , the determinant is .

Combine the opposite terms in .

Subtract from .

Add and .

Can't combine different size matrices.

The combined expressions are .

Factor the characteristic polynomial.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

Factor out the greatest common factor (GCF) from each group.

Factor the polynomial by factoring out the greatest common factor, .

Rewrite as .

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, where and .

Remove unnecessary parentheses.

Set the characteristic polynomial equal to to find the eigenvalues .

If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .

Set the first factor equal to and solve.

Set the first factor equal to .

Subtract from both sides of the equation.

Multiply each term in by

Multiply each term in by .

Multiply .

Multiply by .

Multiply by .

Multiply by .

Set the next factor equal to and solve.

Set the next factor equal to .

Subtract from both sides of the equation.

Set the next factor equal to and solve.

Set the next factor equal to .

Add to both sides of the equation.

The final solution is all the values that make true.

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