Linear Algebra Examples

Find the Eigenvalues [[0.6,0.8],[0.8,-0.6]]
Step 1
Set up the formula to find the characteristic equation .
Step 2
The identity matrix or unit matrix of size is the square matrix with ones on the main diagonal and zeros elsewhere.
Step 3
Substitute the known values into .
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Step 3.1
Substitute for .
Step 3.2
Substitute for .
Step 4
Simplify.
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Step 4.1
Simplify each term.
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Step 4.1.1
Multiply by each element of the matrix.
Step 4.1.2
Simplify each element in the matrix.
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Step 4.1.2.1
Multiply by .
Step 4.1.2.2
Multiply .
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Step 4.1.2.2.1
Multiply by .
Step 4.1.2.2.2
Multiply by .
Step 4.1.2.3
Multiply .
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Step 4.1.2.3.1
Multiply by .
Step 4.1.2.3.2
Multiply by .
Step 4.1.2.4
Multiply by .
Step 4.2
Add the corresponding elements.
Step 4.3
Simplify each element.
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Step 4.3.1
Add and .
Step 4.3.2
Add and .
Step 5
Find the determinant.
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Step 5.1
The determinant of a matrix can be found using the formula .
Step 5.2
Simplify the determinant.
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Step 5.2.1
Simplify each term.
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Step 5.2.1.1
Expand using the FOIL Method.
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Step 5.2.1.1.1
Apply the distributive property.
Step 5.2.1.1.2
Apply the distributive property.
Step 5.2.1.1.3
Apply the distributive property.
Step 5.2.1.2
Simplify and combine like terms.
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Step 5.2.1.2.1
Simplify each term.
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Step 5.2.1.2.1.1
Multiply by .
Step 5.2.1.2.1.2
Multiply by .
Step 5.2.1.2.1.3
Multiply by .
Step 5.2.1.2.1.4
Rewrite using the commutative property of multiplication.
Step 5.2.1.2.1.5
Multiply by by adding the exponents.
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Step 5.2.1.2.1.5.1
Move .
Step 5.2.1.2.1.5.2
Multiply by .
Step 5.2.1.2.1.6
Multiply by .
Step 5.2.1.2.1.7
Multiply by .
Step 5.2.1.2.2
Add and .
Step 5.2.1.3
Multiply by .
Step 5.2.1.4
Add and .
Step 5.2.1.5
Multiply by .
Step 5.2.2
Subtract from .
Step 6
Set the characteristic polynomial equal to to find the eigenvalues .
Step 7
Solve for .
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Step 7.1
Add to both sides of the equation.
Step 7.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 7.3
Simplify .
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Step 7.3.1
Rewrite as .
Step 7.3.2
Pull terms out from under the radical, assuming positive real numbers.
Step 7.4
The complete solution is the result of both the positive and negative portions of the solution.
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Step 7.4.1
First, use the positive value of the to find the first solution.
Step 7.4.2
Next, use the negative value of the to find the second solution.
Step 7.4.3
The complete solution is the result of both the positive and negative portions of the solution.