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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 and combine like terms.
Step 5.2.1.2.1
Simplify each term.
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.
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
Step 7.1
Factor using the AC method.
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 .
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 .
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.