Finite Math Examples

Find the Eigenvalues [[8,6],[4,-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 .
Tap for more steps...
Step 3.1
Substitute for .
Step 3.2
Substitute for .
Step 4
Simplify.
Tap for more steps...
Step 4.1
Simplify each term.
Tap for more steps...
Step 4.1.1
Multiply by each element of the matrix.
Step 4.1.2
Simplify each element in the matrix.
Tap for more steps...
Step 4.1.2.1
Multiply by .
Step 4.1.2.2
Multiply .
Tap for more steps...
Step 4.1.2.2.1
Multiply by .
Step 4.1.2.2.2
Multiply by .
Step 4.1.2.3
Multiply .
Tap for more steps...
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.
Tap for more steps...
Step 4.3.1
Add and .
Step 4.3.2
Add and .
Step 5
Find the determinant.
Tap for more steps...
Step 5.1
The determinant of a matrix can be found using the formula .
Step 5.2
Simplify the determinant.
Tap for more steps...
Step 5.2.1
Simplify each term.
Tap for more steps...
Step 5.2.1.1
Expand using the FOIL Method.
Tap for more steps...
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.
Tap for more steps...
Step 5.2.1.2.1
Simplify each term.
Tap for more steps...
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.
Tap for more steps...
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 .
Tap for more steps...
Step 7.1
Factor using the AC method.
Tap for more steps...
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 .
Tap for more steps...
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 .
Tap for more steps...
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.