Linear Algebra Examples

Find the Eigenvalues [[0,1],[-1, square root of 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
Subtract from .
Step 4.3.2
Add and .
Step 4.3.3
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
Apply the distributive property.
Step 5.2.1.2
Rewrite using the commutative property of multiplication.
Step 5.2.1.3
Simplify each term.
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Step 5.2.1.3.1
Multiply by by adding the exponents.
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Step 5.2.1.3.1.1
Move .
Step 5.2.1.3.1.2
Multiply by .
Step 5.2.1.3.2
Multiply by .
Step 5.2.1.3.3
Multiply by .
Step 5.2.1.4
Multiply .
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Step 5.2.1.4.1
Multiply by .
Step 5.2.1.4.2
Multiply by .
Step 5.2.2
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
Use the quadratic formula to find the solutions.
Step 7.2
Substitute the values , , and into the quadratic formula and solve for .
Step 7.3
Simplify.
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Step 7.3.1
Simplify the numerator.
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Step 7.3.1.1
Apply the product rule to .
Step 7.3.1.2
Raise to the power of .
Step 7.3.1.3
Multiply by .
Step 7.3.1.4
Rewrite as .
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Step 7.3.1.4.1
Use to rewrite as .
Step 7.3.1.4.2
Apply the power rule and multiply exponents, .
Step 7.3.1.4.3
Combine and .
Step 7.3.1.4.4
Cancel the common factor of .
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Step 7.3.1.4.4.1
Cancel the common factor.
Step 7.3.1.4.4.2
Rewrite the expression.
Step 7.3.1.4.5
Evaluate the exponent.
Step 7.3.1.5
Multiply .
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Step 7.3.1.5.1
Multiply by .
Step 7.3.1.5.2
Multiply by .
Step 7.3.1.6
Subtract from .
Step 7.3.1.7
Rewrite as .
Step 7.3.1.8
Rewrite as .
Step 7.3.1.9
Rewrite as .
Step 7.3.2
Multiply by .
Step 7.4
The final answer is the combination of both solutions.