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

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.
Subtract the eigenvalue times the identity matrix from the original matrix.
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Multiply by each element of the matrix.
Simplify each element of the matrix .
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Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Add the corresponding elements of to each element of .
Simplify each element of the matrix .
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Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
The determinant of is .
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Set up the determinant by breaking it into smaller components.
Since the matrix is multiplied by , the determinant is .
The determinant of is .
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The determinant of a matrix can be found using the formula .
Simplify the determinant.
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Simplify terms.
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Simplify each term.
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Expand using the FOIL Method.
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Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
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Simplify each term.
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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.
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Simplify each term.
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Multiply by by adding the exponents.
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Move .
Multiply by .
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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.
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Add and .
Subtract from .
Since the matrix is multiplied by , the determinant is .
Combine the opposite terms in .
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Subtract from .
Add and .
Can't combine different size matrices.
The combined expressions are .
Factor the characteristic polynomial.
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Factor out the greatest common factor from each group.
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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.
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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 .
Solve the equation for .
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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.
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Set the first factor equal to .
Subtract from both sides of the equation.
Multiply each term in by
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Multiply each term in by .
Multiply .
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Multiply by .
Multiply by .
Multiply by .
Set the next factor equal to and solve.
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Set the next factor equal to .
Subtract from both sides of the equation.
Set the next factor equal to and solve.
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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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