Linear Algebra Examples

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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Simplify element by multiplying .
Simplify element by multiplying .
Simplify element by multiplying .
Simplify element by multiplying .
Combine the similar matrices with each others.
Simplify each element of the matrix .
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Combine the same size matrices and by adding the corresponding elements of each.
Simplify element of the matrix.
Simplify element of the matrix.
The determinant of is .
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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.
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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.
Remove parentheses.
Simplify and combine like terms.
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Simplify each term.
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Multiply by by adding the exponents.
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Move .
Combine and
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Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Simplify .
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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 .
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Use the quadratic formula to find the solutions.
Substitute the values , , and into the quadratic formula and solve for .
Simplify.
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Simplify the numerator.
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Raise to the power of .
Multiply by .
Multiply by .
Add and .
Rewrite as .
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Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Simplify .
The final answer is the combination of both solutions.
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