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Finite Math Examples
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
Step 3.1
Substitute for .
Step 3.2
Substitute for .
Step 4
Step 4.1
Simplify each term.
Step 4.1.1
Multiply by each element of the matrix.
Step 4.1.2
Simplify each element in the matrix.
Step 4.1.2.1
Multiply by .
Step 4.1.2.2
Multiply .
Step 4.1.2.2.1
Multiply by .
Step 4.1.2.2.2
Multiply by .
Step 4.1.2.3
Multiply .
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.
Step 4.3.1
Add and .
Step 4.3.2
Add and .
Step 5
Step 5.1
The determinant of a matrix can be found using the formula .
Step 5.2
Simplify the determinant.
Step 5.2.1
Simplify each term.
Step 5.2.1.1
Expand using the FOIL Method.
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 each term.
Step 5.2.1.2.1
Move to the left of .
Step 5.2.1.2.2
Rewrite using the commutative property of multiplication.
Step 5.2.1.2.3
Multiply by .
Step 5.2.1.2.4
Rewrite using the commutative property of multiplication.
Step 5.2.1.2.5
Multiply by by adding the exponents.
Step 5.2.1.2.5.1
Move .
Step 5.2.1.2.5.2
Multiply by .
Step 5.2.1.2.6
Multiply by .
Step 5.2.1.2.7
Multiply by .
Step 5.2.1.3
Multiply by .
Step 5.2.2
Move .
Step 5.2.3
Move .
Step 6
Set the characteristic polynomial equal to to find the eigenvalues .
Step 7
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.
Step 7.3.1
Simplify the numerator.
Step 7.3.1.1
Apply the distributive property.
Step 7.3.1.2
Multiply .
Step 7.3.1.2.1
Multiply by .
Step 7.3.1.2.2
Multiply by .
Step 7.3.1.3
Multiply by .
Step 7.3.1.4
Rewrite as .
Step 7.3.1.5
Expand using the FOIL Method.
Step 7.3.1.5.1
Apply the distributive property.
Step 7.3.1.5.2
Apply the distributive property.
Step 7.3.1.5.3
Apply the distributive property.
Step 7.3.1.6
Simplify and combine like terms.
Step 7.3.1.6.1
Simplify each term.
Step 7.3.1.6.1.1
Rewrite using the commutative property of multiplication.
Step 7.3.1.6.1.2
Multiply by by adding the exponents.
Step 7.3.1.6.1.2.1
Move .
Step 7.3.1.6.1.2.2
Multiply by .
Step 7.3.1.6.1.3
Multiply by .
Step 7.3.1.6.1.4
Multiply by .
Step 7.3.1.6.1.5
Multiply by .
Step 7.3.1.6.1.6
Multiply by .
Step 7.3.1.6.1.7
Multiply by .
Step 7.3.1.6.2
Add and .
Step 7.3.1.7
Multiply by .
Step 7.3.1.8
Apply the distributive property.
Step 7.3.1.9
Multiply by .
Step 7.3.1.10
Multiply by .
Step 7.3.1.11
Subtract from .
Step 7.3.1.12
Add and .
Step 7.3.2
Multiply by .
Step 7.4
The final answer is the combination of both solutions.