Linear Algebra 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
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 .
Tap for more steps...
Step 4.1.2.4.1
Multiply by .
Step 4.1.2.4.2
Multiply by .
Step 4.1.2.5
Multiply by .
Step 4.1.2.6
Multiply .
Tap for more steps...
Step 4.1.2.6.1
Multiply by .
Step 4.1.2.6.2
Multiply by .
Step 4.1.2.7
Multiply .
Tap for more steps...
Step 4.1.2.7.1
Multiply by .
Step 4.1.2.7.2
Multiply by .
Step 4.1.2.8
Multiply .
Tap for more steps...
Step 4.1.2.8.1
Multiply by .
Step 4.1.2.8.2
Multiply by .
Step 4.1.2.9
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 4.3.3
Add and .
Step 4.3.4
Subtract from .
Step 4.3.5
Add and .
Step 4.3.6
Add and .
Step 4.3.7
Add and .
Step 5
Find the determinant.
Tap for more steps...
Step 5.1
Choose the row or column with the most elements. If there are no elements choose any row or column. Multiply every element in column by its cofactor and add.
Tap for more steps...
Step 5.1.1
Consider the corresponding sign chart.
Step 5.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 5.1.3
The minor for is the determinant with row and column deleted.
Step 5.1.4
Multiply element by its cofactor.
Step 5.1.5
The minor for is the determinant with row and column deleted.
Step 5.1.6
Multiply element by its cofactor.
Step 5.1.7
The minor for is the determinant with row and column deleted.
Step 5.1.8
Multiply element by its cofactor.
Step 5.1.9
Add the terms together.
Step 5.2
Multiply by .
Step 5.3
Evaluate .
Tap for more steps...
Step 5.3.1
The determinant of a matrix can be found using the formula .
Step 5.3.2
Simplify the determinant.
Tap for more steps...
Step 5.3.2.1
Simplify each term.
Tap for more steps...
Step 5.3.2.1.1
Apply the distributive property.
Step 5.3.2.1.2
Multiply by .
Step 5.3.2.1.3
Rewrite using the commutative property of multiplication.
Step 5.3.2.1.4
Simplify each term.
Tap for more steps...
Step 5.3.2.1.4.1
Multiply by by adding the exponents.
Tap for more steps...
Step 5.3.2.1.4.1.1
Move .
Step 5.3.2.1.4.1.2
Multiply by .
Step 5.3.2.1.4.2
Multiply by .
Step 5.3.2.1.4.3
Multiply by .
Step 5.3.2.1.5
Multiply by .
Step 5.3.2.2
Add and .
Step 5.3.2.3
Reorder and .
Step 5.4
Evaluate .
Tap for more steps...
Step 5.4.1
The determinant of a matrix can be found using the formula .
Step 5.4.2
Simplify the determinant.
Tap for more steps...
Step 5.4.2.1
Simplify each term.
Tap for more steps...
Step 5.4.2.1.1
Apply the distributive property.
Step 5.4.2.1.2
Multiply by .
Step 5.4.2.1.3
Multiply by .
Step 5.4.2.1.4
Multiply by .
Step 5.4.2.2
Combine the opposite terms in .
Tap for more steps...
Step 5.4.2.2.1
Subtract from .
Step 5.4.2.2.2
Add and .
Step 5.5
Simplify the determinant.
Tap for more steps...
Step 5.5.1
Add and .
Step 5.5.2
Simplify each term.
Tap for more steps...
Step 5.5.2.1
Expand using the FOIL Method.
Tap for more steps...
Step 5.5.2.1.1
Apply the distributive property.
Step 5.5.2.1.2
Apply the distributive property.
Step 5.5.2.1.3
Apply the distributive property.
Step 5.5.2.2
Simplify and combine like terms.
Tap for more steps...
Step 5.5.2.2.1
Simplify each term.
Tap for more steps...
Step 5.5.2.2.1.1
Multiply by .
Step 5.5.2.2.1.2
Multiply by by adding the exponents.
Tap for more steps...
Step 5.5.2.2.1.2.1
Move .
Step 5.5.2.2.1.2.2
Multiply by .
Tap for more steps...
Step 5.5.2.2.1.2.2.1
Raise to the power of .
Step 5.5.2.2.1.2.2.2
Use the power rule to combine exponents.
Step 5.5.2.2.1.2.3
Add and .
Step 5.5.2.2.1.3
Rewrite using the commutative property of multiplication.
Step 5.5.2.2.1.4
Multiply by by adding the exponents.
Tap for more steps...
Step 5.5.2.2.1.4.1
Move .
Step 5.5.2.2.1.4.2
Multiply by .
Step 5.5.2.2.1.5
Multiply by .
Step 5.5.2.2.1.6
Multiply by .
Step 5.5.2.2.2
Add and .
Step 5.5.2.3
Multiply by .
Step 5.5.3
Combine the opposite terms in .
Tap for more steps...
Step 5.5.3.1
Add and .
Step 5.5.3.2
Add and .
Step 5.5.4
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 out of .
Tap for more steps...
Step 7.1.1
Factor out of .
Step 7.1.2
Factor out of .
Step 7.1.3
Factor out of .
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
Solve for .
Tap for more steps...
Step 7.3.2.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 7.3.2.2
Simplify .
Tap for more steps...
Step 7.3.2.2.1
Rewrite as .
Step 7.3.2.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 7.3.2.2.3
Plus or minus is .
Step 7.4
Set equal to and solve for .
Tap for more steps...
Step 7.4.1
Set equal to .
Step 7.4.2
Add to both sides of the equation.
Step 7.5
The final solution is all the values that make true.
Enter YOUR Problem
Mathway requires javascript and a modern browser.