Linear Algebra Examples

Step 1
Find the eigenvalues.
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Step 1.1
Set up the formula to find the characteristic equation .
Step 1.2
The identity matrix or unit matrix of size is the square matrix with ones on the main diagonal and zeros elsewhere.
Step 1.3
Substitute the known values into .
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Step 1.3.1
Substitute for .
Step 1.3.2
Substitute for .
Step 1.4
Simplify.
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Step 1.4.1
Simplify each term.
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Step 1.4.1.1
Multiply by each element of the matrix.
Step 1.4.1.2
Simplify each element in the matrix.
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Step 1.4.1.2.1
Multiply by .
Step 1.4.1.2.2
Multiply .
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Step 1.4.1.2.2.1
Multiply by .
Step 1.4.1.2.2.2
Multiply by .
Step 1.4.1.2.3
Multiply .
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Step 1.4.1.2.3.1
Multiply by .
Step 1.4.1.2.3.2
Multiply by .
Step 1.4.1.2.4
Multiply by .
Step 1.4.2
Add the corresponding elements.
Step 1.4.3
Simplify each element.
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Step 1.4.3.1
Add and .
Step 1.4.3.2
Add and .
Step 1.5
Find the determinant.
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Step 1.5.1
The determinant of a matrix can be found using the formula .
Step 1.5.2
Simplify the determinant.
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Step 1.5.2.1
Simplify each term.
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Step 1.5.2.1.1
Expand using the FOIL Method.
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Step 1.5.2.1.1.1
Apply the distributive property.
Step 1.5.2.1.1.2
Apply the distributive property.
Step 1.5.2.1.1.3
Apply the distributive property.
Step 1.5.2.1.2
Simplify and combine like terms.
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Step 1.5.2.1.2.1
Simplify each term.
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Step 1.5.2.1.2.1.1
Multiply by .
Step 1.5.2.1.2.1.2
Multiply by .
Step 1.5.2.1.2.1.3
Multiply by .
Step 1.5.2.1.2.1.4
Rewrite using the commutative property of multiplication.
Step 1.5.2.1.2.1.5
Multiply by by adding the exponents.
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Step 1.5.2.1.2.1.5.1
Move .
Step 1.5.2.1.2.1.5.2
Multiply by .
Step 1.5.2.1.2.1.6
Multiply by .
Step 1.5.2.1.2.1.7
Multiply by .
Step 1.5.2.1.2.2
Subtract from .
Step 1.5.2.1.3
Multiply by .
Step 1.5.2.2
Subtract from .
Step 1.5.2.3
Reorder and .
Step 1.6
Set the characteristic polynomial equal to to find the eigenvalues .
Step 1.7
Solve for .
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Step 1.7.1
Use the quadratic formula to find the solutions.
Step 1.7.2
Substitute the values , , and into the quadratic formula and solve for .
Step 1.7.3
Simplify.
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Step 1.7.3.1
Simplify the numerator.
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Step 1.7.3.1.1
Raise to the power of .
Step 1.7.3.1.2
Multiply .
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Step 1.7.3.1.2.1
Multiply by .
Step 1.7.3.1.2.2
Multiply by .
Step 1.7.3.1.3
Add and .
Step 1.7.3.2
Multiply by .
Step 1.7.4
The final answer is the combination of both solutions.
Step 2
The eigenvector is equal to the null space of the matrix minus the eigenvalue times the identity matrix where is the null space and is the identity matrix.
Step 3
Find the eigenvector using the eigenvalue .
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Step 3.1
Substitute the known values into the formula.
Step 3.2
Simplify.
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Step 3.2.1
Simplify each term.
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Step 3.2.1.1
Multiply by each element of the matrix.
Step 3.2.1.2
Simplify each element in the matrix.
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Step 3.2.1.2.1
Multiply by .
Step 3.2.1.2.2
Multiply .
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Step 3.2.1.2.2.1
Multiply by .
Step 3.2.1.2.2.2
Multiply by .
Step 3.2.1.2.3
Multiply .
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Step 3.2.1.2.3.1
Multiply by .
Step 3.2.1.2.3.2
Multiply by .
Step 3.2.1.2.4
Multiply by .
Step 3.2.2
Add the corresponding elements.
Step 3.2.3
Simplify each element.
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Step 3.2.3.1
Write as a fraction with a common denominator.
Step 3.2.3.2
Combine the numerators over the common denominator.
Step 3.2.3.3
Simplify the numerator.
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Step 3.2.3.3.1
Apply the distributive property.
Step 3.2.3.3.2
Multiply by .
Step 3.2.3.3.3
Subtract from .
Step 3.2.3.4
Rewrite as .
Step 3.2.3.5
Factor out of .
Step 3.2.3.6
Factor out of .
Step 3.2.3.7
Move the negative in front of the fraction.
Step 3.2.3.8
Add and .
Step 3.2.3.9
Add and .
Step 3.2.3.10
To write as a fraction with a common denominator, multiply by .
Step 3.2.3.11
Combine and .
Step 3.2.3.12
Combine the numerators over the common denominator.
Step 3.2.3.13
Simplify the numerator.
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Step 3.2.3.13.1
Multiply by .
Step 3.2.3.13.2
Apply the distributive property.
Step 3.2.3.13.3
Multiply by .
Step 3.2.3.13.4
Subtract from .
Step 3.3
Find the null space when .
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Step 3.3.1
Write as an augmented matrix for .
Step 3.3.2
Find the reduced row echelon form.
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Step 3.3.2.1
Multiply each element of by to make the entry at a .
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Step 3.3.2.1.1
Multiply each element of by to make the entry at a .
Step 3.3.2.1.2
Simplify .
Step 3.3.2.2
Perform the row operation to make the entry at a .
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Step 3.3.2.2.1
Perform the row operation to make the entry at a .
Step 3.3.2.2.2
Simplify .
Step 3.3.3
Use the result matrix to declare the final solution to the system of equations.
Step 3.3.4
Write a solution vector by solving in terms of the free variables in each row.
Step 3.3.5
Write the solution as a linear combination of vectors.
Step 3.3.6
Write as a solution set.
Step 3.3.7
The solution is the set of vectors created from the free variables of the system.
Step 4
Find the eigenvector using the eigenvalue .
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Step 4.1
Substitute the known values into the formula.
Step 4.2
Simplify.
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Step 4.2.1
Simplify each term.
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Step 4.2.1.1
Multiply by each element of the matrix.
Step 4.2.1.2
Simplify each element in the matrix.
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Step 4.2.1.2.1
Multiply by .
Step 4.2.1.2.2
Multiply .
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Step 4.2.1.2.2.1
Multiply by .
Step 4.2.1.2.2.2
Multiply by .
Step 4.2.1.2.3
Multiply .
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Step 4.2.1.2.3.1
Multiply by .
Step 4.2.1.2.3.2
Multiply by .
Step 4.2.1.2.4
Multiply by .
Step 4.2.2
Add the corresponding elements.
Step 4.2.3
Simplify each element.
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Step 4.2.3.1
Write as a fraction with a common denominator.
Step 4.2.3.2
Combine the numerators over the common denominator.
Step 4.2.3.3
Simplify the numerator.
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Step 4.2.3.3.1
Apply the distributive property.
Step 4.2.3.3.2
Multiply by .
Step 4.2.3.3.3
Multiply .
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Step 4.2.3.3.3.1
Multiply by .
Step 4.2.3.3.3.2
Multiply by .
Step 4.2.3.3.4
Subtract from .
Step 4.2.3.4
Rewrite as .
Step 4.2.3.5
Factor out of .
Step 4.2.3.6
Factor out of .
Step 4.2.3.7
Move the negative in front of the fraction.
Step 4.2.3.8
Add and .
Step 4.2.3.9
Add and .
Step 4.2.3.10
To write as a fraction with a common denominator, multiply by .
Step 4.2.3.11
Combine and .
Step 4.2.3.12
Combine the numerators over the common denominator.
Step 4.2.3.13
Simplify the numerator.
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Step 4.2.3.13.1
Multiply by .
Step 4.2.3.13.2
Apply the distributive property.
Step 4.2.3.13.3
Multiply by .
Step 4.2.3.13.4
Multiply .
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Step 4.2.3.13.4.1
Multiply by .
Step 4.2.3.13.4.2
Multiply by .
Step 4.2.3.13.5
Subtract from .
Step 4.3
Find the null space when .
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Step 4.3.1
Write as an augmented matrix for .
Step 4.3.2
Find the reduced row echelon form.
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Step 4.3.2.1
Multiply each element of by to make the entry at a .
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Step 4.3.2.1.1
Multiply each element of by to make the entry at a .
Step 4.3.2.1.2
Simplify .
Step 4.3.2.2
Perform the row operation to make the entry at a .
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Step 4.3.2.2.1
Perform the row operation to make the entry at a .
Step 4.3.2.2.2
Simplify .
Step 4.3.3
Use the result matrix to declare the final solution to the system of equations.
Step 4.3.4
Write a solution vector by solving in terms of the free variables in each row.
Step 4.3.5
Write the solution as a linear combination of vectors.
Step 4.3.6
Write as a solution set.
Step 4.3.7
The solution is the set of vectors created from the free variables of the system.
Step 5
The eigenspace of is the list of the vector space for each eigenvalue.
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