Linear Algebra Examples

Set up the formula to find the characteristic equation .
Substitute the known values in the formula.
Subtract the eigenvalue times the identity matrix from the original matrix.
Tap for more steps...
Multiply by each element of the matrix.
Simplify each element of the matrix .
Tap for more steps...
Simplify element by multiplying to get .
Simplify element by multiplying to get .
Simplify element by multiplying to get .
Simplify element by multiplying to get .
Simplify element by multiplying to get .
Simplify element by multiplying to get .
Simplify element by multiplying to get .
Simplify element by multiplying to get .
Simplify element by multiplying to get .
Combine the similar matrices with each others.
Simplify each element of the matrix .
Tap for more steps...
Combine the same size matrices and by adding the corresponding elements of each.
Simplify element of the matrix.
Simplify element of the matrix.
Simplify element of the matrix.
Simplify element of the matrix.
Simplify element of the matrix.
Simplify element of the matrix.
The determinant of is .
Tap for more steps...
Set up the determinant by breaking it into smaller components.
The determinant of is .
Tap for more steps...
The determinant of a matrix can be found using the formula .
Simplify the determinant.
Tap for more steps...
Simplify each term.
Tap for more steps...
Multiply by to get .
Multiply by to get .
Simplify by multiplying through.
Tap for more steps...
Add and to get .
Apply the distributive property.
Multiply.
Tap for more steps...
Multiply by to get .
Multiply by to get .
The determinant of is .
Tap for more steps...
The determinant of a matrix can be found using the formula .
Simplify the determinant.
Tap for more steps...
Simplify terms.
Tap for more steps...
Simplify each term.
Tap for more steps...
Expand using the FOIL Method.
Tap for more steps...
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Remove parentheses.
Simplify and combine like terms.
Tap for more steps...
Simplify each term.
Tap for more steps...
Move .
Use the power rule to combine exponents.
Add and to get .
Simplify .
Tap for more steps...
Multiply by to get .
Multiply by to get .
Simplify .
Tap for more steps...
Multiply by to get .
Multiply by to get .
Simplify .
Tap for more steps...
Multiply by to get .
Multiply by to get .
Multiply by to get .
Add and to get .
Multiply by to get .
Add and to get .
Expand by multiplying each term in the first expression by each term in the second expression.
Simplify terms.
Tap for more steps...
Remove unnecessary parentheses.
Simplify each term.
Tap for more steps...
Move .
Use the power rule to combine exponents.
Add and to get .
Move .
Use the power rule to combine exponents.
Add and to get .
Multiply by to get .
Multiply by to get .
Multiply by to get .
Multiply by to get .
Multiply by to get .
Simplify by adding terms.
Tap for more steps...
Add and to get .
Add and to get .
Since the matrix is multiplied by , the determinant is .
Subtract from to get .
Simplify by adding numbers.
Tap for more steps...
Add and to get .
Simplify by adding zeros.
Tap for more steps...
Add and to get .
Add and to get .
Factor the characteristic polynomial.
Tap for more steps...
Factor out of .
Tap for more steps...
Factor out of .
Tap for more steps...
Factor out of .
Move .
Multiply by to get .
Factor out of .
Tap for more steps...
Factor out of .
Move .
Multiply by to get .
Reorder and .
Factor out of .
Factor out of .
Factor.
Tap for more steps...
Factor by grouping.
Tap for more steps...
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Tap for more steps...
Factor out of .
Rewrite as plus
Apply the distributive property.
Multiply by to get .
Remove parentheses.
Factor out the greatest common factor from each group.
Tap for more steps...
Group the first two terms and the last two terms.
Factor out the greatest common factor (GCF) from each group.
Factor the polynomial by factoring out the greatest common factor, .
Remove unnecessary parentheses.
Set the characteristic polynomial equal to to find the eigenvalues .
Solve the equation for .
Tap for more steps...
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Set the first factor equal to .
Set the next factor equal to and solve.
Tap for more steps...
Set the next factor equal to .
Since does not contain the variable to solve for, move it to the right side of the equation by adding to both sides.
Multiply each term in by
Tap for more steps...
Multiply each term in by .
Simplify .
Tap for more steps...
Multiply by to get .
Multiply by to get .
Multiply by to get .
Set the next factor equal to and solve.
Tap for more steps...
Set the next factor equal to .
Since does not contain the variable to solve for, move it to the right side of the equation by adding to both sides.
The final solution is all the values that make true.
Enter YOUR Problem

Enter the email address associated with your Mathway account below and we'll send you a link to reset your password.

Please enter an email address
Please enter a valid email address
The email address you entered was not found in our system
The email address you entered is associated with a Facebook user
We're sorry, we were unable to process your request at this time

Mathway Premium

Step-by-step work + explanations
  •    Step-by-step work
  •    Detailed explanations
  •    No advertisements
  •    Access anywhere
Access the steps on both the Mathway website and mobile apps
$--.--/month
$--.--/year (--%)

Mathway Premium

Visa and MasterCard security codes are located on the back of card and are typically a separate group of 3 digits to the right of the signature strip.

American Express security codes are 4 digits located on the front of the card and usually towards the right.
This option is required to subscribe.
Go Back

Step-by-step upgrade complete!

Mathway requires javascript and a modern browser.
  [ x 2     1 2     π     x d x   ]