Linear Algebra Examples

Step 1
Set up the formula to find the characteristic equation .
Step 2
Substitute the known values in the formula.
Step 3
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 in the matrix.
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Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Add the corresponding elements.
Simplify each element of the matrix .
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Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Step 4
Find the determinant of .
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Set up the determinant by breaking it into smaller components.
Find the determinant of .
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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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Multiply by .
Multiply by .
Simplify by multiplying through.
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Add and .
Apply the distributive property.
Multiply.
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Multiply by .
Multiply by .
Find the determinant of .
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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 .
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Multiply by .
Multiply by .
Multiply .
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Multiply by .
Multiply by .
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
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Move .
Multiply by .
Multiply by .
Multiply by .
Add and .
Multiply by .
Add and .
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 .
Multiply by .
Multiply by .
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
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Move .
Multiply by .
Multiply by .
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 .
Simplify by adding terms.
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Subtract from .
Subtract from .
Since the matrix is multiplied by , the determinant is .
Combine the opposite terms in .
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Add and .
Add and .
Add and .
Subtract from .
Factor the characteristic polynomial.
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Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Let . Substitute for all occurrences of .
Factor by grouping.
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Reorder terms.
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
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Factor out of .
Rewrite as plus
Apply the distributive property.
Multiply by .
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, .
Factor.
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Replace all occurrences of with .
Remove unnecessary parentheses.
Step 5
Set the characteristic polynomial equal to to find the eigenvalues .
Step 6
Find the roots of by solving 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 equal to .
Set equal to and solve for .
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Set equal to .
Solve for .
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Subtract from both sides of the equation.
Divide each term in by and simplify.
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Divide each term in by .
Simplify the left side.
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Dividing two negative values results in a positive value.
Divide by .
Simplify the right side.
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Divide by .
Set equal to and solve for .
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Set equal to .
Subtract from both sides of the equation.
The final solution is all the values that make true.
Step 7
Enter YOUR Problem
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