Linear Algebra Examples

Find the Eigenvalues [[1,4],[1,-2]]
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 5.2.2
Subtract from .
Step 5.2.3
Reorder and .
Step 6
Set the characteristic polynomial equal to to find the eigenvalues .
Step 7
Solve for .
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Step 7.1
Factor using the AC method.
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Step 7.1.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 7.1.2
Write the factored form using these integers.
Step 7.2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 7.3
Set equal to and solve for .
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Step 7.3.1
Set equal to .
Step 7.3.2
Add to both sides of the equation.
Step 7.4
Set equal to and solve for .
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Step 7.4.1
Set equal to .
Step 7.4.2
Subtract from both sides of the equation.
Step 7.5
The final solution is all the values that make true.