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 .
Step 3.1
Substitute for .
Step 3.2
Substitute for .
Step 4
Simplify.
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 .
Step 4.1.2.4.1
Multiply by .
Step 4.1.2.4.2
Multiply by .
Step 4.1.2.5
Multiply .
Step 4.1.2.5.1
Multiply by .
Step 4.1.2.5.2
Multiply by .
Step 4.1.2.6
Multiply by .
Step 4.1.2.7
Multiply .
Step 4.1.2.7.1
Multiply by .
Step 4.1.2.7.2
Multiply by .
Step 4.1.2.8
Multiply .
Step 4.1.2.8.1
Multiply by .
Step 4.1.2.8.2
Multiply by .
Step 4.1.2.9
Multiply .
Step 4.1.2.9.1
Multiply by .
Step 4.1.2.9.2
Multiply by .
Step 4.1.2.10
Multiply .
Step 4.1.2.10.1
Multiply by .
Step 4.1.2.10.2
Multiply by .
Step 4.1.2.11
Multiply by .
Step 4.1.2.12
Multiply .
Step 4.1.2.12.1
Multiply by .
Step 4.1.2.12.2
Multiply by .
Step 4.1.2.13
Multiply .
Step 4.1.2.13.1
Multiply by .
Step 4.1.2.13.2
Multiply by .
Step 4.1.2.14
Multiply .
Step 4.1.2.14.1
Multiply by .
Step 4.1.2.14.2
Multiply by .
Step 4.1.2.15
Multiply .
Step 4.1.2.15.1
Multiply by .
Step 4.1.2.15.2
Multiply by .
Step 4.1.2.16
Multiply by .
Step 4.2
Step 4.3
Simplify each element.
Step 4.3.1
Step 4.3.2
Step 4.3.3
Step 4.3.4
Step 4.3.5
Step 4.3.6
Step 4.3.7
Step 4.3.8
Step 4.3.9
Step 4.3.10
Step 4.3.11
Step 4.3.12
Step 5
Find the determinant.
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 row by its cofactor and add.
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
The minor for is the determinant with row and column deleted.
Step 5.1.10
Multiply element by its cofactor.
Step 5.1.11
Step 5.2
Evaluate .
Step 5.2.1
Choose the row or column with the most elements. If there are no elements choose any row or column. Multiply every element in row by its cofactor and add.
Step 5.2.1.1
Consider the corresponding sign chart.
Step 5.2.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 5.2.1.3
The minor for is the determinant with row and column deleted.
Step 5.2.1.4
Multiply element by its cofactor.
Step 5.2.1.5
The minor for is the determinant with row and column deleted.
Step 5.2.1.6
Multiply element by its cofactor.
Step 5.2.1.7
The minor for is the determinant with row and column deleted.
Step 5.2.1.8
Multiply element by its cofactor.
Step 5.2.1.9
Step 5.2.2
Evaluate .
Step 5.2.2.1
The determinant of a matrix can be found using the formula .
Step 5.2.2.2
Simplify the determinant.
Step 5.2.2.2.1
Simplify each term.
Step 5.2.2.2.1.1
Expand using the FOIL Method.
Step 5.2.2.2.1.1.1
Apply the distributive property.
Step 5.2.2.2.1.1.2
Apply the distributive property.
Step 5.2.2.2.1.1.3
Apply the distributive property.
Step 5.2.2.2.1.2
Simplify and combine like terms.
Step 5.2.2.2.1.2.1
Simplify each term.
Step 5.2.2.2.1.2.1.1
Multiply by .
Step 5.2.2.2.1.2.1.2
Multiply by .
Step 5.2.2.2.1.2.1.3
Multiply by .
Step 5.2.2.2.1.2.1.4
Rewrite using the commutative property of multiplication.
Step 5.2.2.2.1.2.1.5
Multiply by by adding the exponents.
Step 5.2.2.2.1.2.1.5.1
Move .
Step 5.2.2.2.1.2.1.5.2
Multiply by .
Step 5.2.2.2.1.2.1.6
Multiply by .
Step 5.2.2.2.1.2.1.7
Multiply by .
Step 5.2.2.2.1.2.2
Subtract from .
Step 5.2.2.2.1.3
Multiply by .
Step 5.2.2.2.2
Combine the opposite terms in .
Step 5.2.2.2.2.1
Step 5.2.2.2.2.2
Step 5.2.2.2.3
Reorder and .
Step 5.2.3
Evaluate .
Step 5.2.3.1
The determinant of a matrix can be found using the formula .
Step 5.2.3.2
Simplify the determinant.
Step 5.2.3.2.1
Simplify each term.
Step 5.2.3.2.1.1
Apply the distributive property.
Step 5.2.3.2.1.2
Multiply by .
Step 5.2.3.2.1.3
Multiply by .
Step 5.2.3.2.1.4
Multiply .
Step 5.2.3.2.1.4.1
Multiply by .
Step 5.2.3.2.1.4.2
Multiply by .
Step 5.2.3.2.2
Subtract from .
Step 5.2.4
Evaluate .
Step 5.2.4.1
The determinant of a matrix can be found using the formula .
Step 5.2.4.2
Simplify the determinant.
Step 5.2.4.2.1
Simplify each term.
Step 5.2.4.2.1.1
Multiply by .
Step 5.2.4.2.1.2
Apply the distributive property.
Step 5.2.4.2.1.3
Multiply by .
Step 5.2.4.2.1.4
Multiply by .
Step 5.2.4.2.1.5
Apply the distributive property.
Step 5.2.4.2.1.6
Multiply by .
Step 5.2.4.2.1.7
Multiply by .
Step 5.2.4.2.2
Subtract from .
Step 5.2.4.2.3
Reorder and .
Step 5.2.5
Simplify the determinant.
Step 5.2.5.1
Simplify each term.
Step 5.2.5.1.1
Expand using the FOIL Method.
Step 5.2.5.1.1.1
Apply the distributive property.
Step 5.2.5.1.1.2
Apply the distributive property.
Step 5.2.5.1.1.3
Apply the distributive property.
Step 5.2.5.1.2
Simplify and combine like terms.
Step 5.2.5.1.2.1
Simplify each term.
Step 5.2.5.1.2.1.1
Multiply by .
Step 5.2.5.1.2.1.2
Multiply by .
Step 5.2.5.1.2.1.3
Multiply by by adding the exponents.
Step 5.2.5.1.2.1.3.1
Move .
Step 5.2.5.1.2.1.3.2
Multiply by .
Step 5.2.5.1.2.1.3.2.1
Raise to the power of .
Step 5.2.5.1.2.1.3.2.2
Use the power rule to combine exponents.
Step 5.2.5.1.2.1.3.3
Step 5.2.5.1.2.1.4
Rewrite using the commutative property of multiplication.
Step 5.2.5.1.2.1.5
Multiply by by adding the exponents.
Step 5.2.5.1.2.1.5.1
Move .
Step 5.2.5.1.2.1.5.2
Multiply by .
Step 5.2.5.1.2.1.6
Multiply by .
Step 5.2.5.1.2.2
Step 5.2.5.1.3
Apply the distributive property.
Step 5.2.5.1.4
Multiply by .
Step 5.2.5.1.5
Multiply by .
Step 5.2.5.1.6
Apply the distributive property.
Step 5.2.5.1.7
Multiply by .
Step 5.2.5.1.8
Multiply by .
Step 5.2.5.2
Step 5.2.5.3
Subtract from .
Step 5.2.5.4
Step 5.2.5.5
Reorder and .
Step 5.3
Evaluate .
Step 5.3.1
Choose the row or column with the most elements. If there are no elements choose any row or column. Multiply every element in row by its cofactor and add.
Step 5.3.1.1
Consider the corresponding sign chart.
Step 5.3.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 5.3.1.3
The minor for is the determinant with row and column deleted.
Step 5.3.1.4
Multiply element by its cofactor.
Step 5.3.1.5
The minor for is the determinant with row and column deleted.
Step 5.3.1.6
Multiply element by its cofactor.
Step 5.3.1.7
The minor for is the determinant with row and column deleted.
Step 5.3.1.8
Multiply element by its cofactor.
Step 5.3.1.9
Step 5.3.2
Evaluate .
Step 5.3.2.1
The determinant of a matrix can be found using the formula .
Step 5.3.2.2
Simplify the determinant.
Step 5.3.2.2.1
Simplify each term.
Step 5.3.2.2.1.1
Apply the distributive property.
Step 5.3.2.2.1.2
Multiply by .
Step 5.3.2.2.1.3
Multiply by .
Step 5.3.2.2.1.4
Multiply by .
Step 5.3.2.2.2
Subtract from .
Step 5.3.3
Evaluate .
Step 5.3.3.1
The determinant of a matrix can be found using the formula .
Step 5.3.3.2
Simplify the determinant.
Step 5.3.3.2.1
Simplify each term.
Step 5.3.3.2.1.1
Apply the distributive property.
Step 5.3.3.2.1.2
Multiply by .
Step 5.3.3.2.1.3
Multiply by .
Step 5.3.3.2.1.4
Multiply by .
Step 5.3.3.2.2
Subtract from .
Step 5.3.4
Evaluate .
Step 5.3.4.1
The determinant of a matrix can be found using the formula .
Step 5.3.4.2
Simplify the determinant.
Step 5.3.4.2.1
Simplify each term.
Step 5.3.4.2.1.1
Multiply by .
Step 5.3.4.2.1.2
Multiply by .
Step 5.3.4.2.2
Subtract from .
Step 5.3.5
Simplify the determinant.
Step 5.3.5.1
Simplify each term.
Step 5.3.5.1.1
Apply the distributive property.
Step 5.3.5.1.2
Multiply by .
Step 5.3.5.1.3
Multiply by .
Step 5.3.5.1.4
Expand using the FOIL Method.
Step 5.3.5.1.4.1
Apply the distributive property.
Step 5.3.5.1.4.2
Apply the distributive property.
Step 5.3.5.1.4.3
Apply the distributive property.
Step 5.3.5.1.5
Simplify and combine like terms.
Step 5.3.5.1.5.1
Simplify each term.
Step 5.3.5.1.5.1.1
Multiply by .
Step 5.3.5.1.5.1.2
Multiply by .
Step 5.3.5.1.5.1.3
Rewrite using the commutative property of multiplication.
Step 5.3.5.1.5.1.4
Multiply by by adding the exponents.
Step 5.3.5.1.5.1.4.1
Move .
Step 5.3.5.1.5.1.4.2
Multiply by .
Step 5.3.5.1.5.1.5
Multiply by .
Step 5.3.5.1.5.1.6
Multiply by .
Step 5.3.5.1.5.2
Subtract from .
Step 5.3.5.1.6
Multiply by .
Step 5.3.5.2
Subtract from .
Step 5.3.5.3
Subtract from .
Step 5.3.5.4
Step 5.3.5.5
Reorder and .
Step 5.4
Evaluate .
Step 5.4.1
Choose the row or column with the most elements. If there are no elements choose any row or column. Multiply every element in row by its cofactor and add.
Step 5.4.1.1
Consider the corresponding sign chart.
Step 5.4.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 5.4.1.3
The minor for is the determinant with row and column deleted.
Step 5.4.1.4
Multiply element by its cofactor.
Step 5.4.1.5
The minor for is the determinant with row and column deleted.
Step 5.4.1.6
Multiply element by its cofactor.
Step 5.4.1.7
The minor for is the determinant with row and column deleted.
Step 5.4.1.8
Multiply element by its cofactor.
Step 5.4.1.9
Step 5.4.2
Evaluate .
Step 5.4.2.1
The determinant of a matrix can be found using the formula .
Step 5.4.2.2
Simplify the determinant.
Step 5.4.2.2.1
Simplify each term.
Step 5.4.2.2.1.1
Apply the distributive property.
Step 5.4.2.2.1.2
Multiply by .
Step 5.4.2.2.1.3
Multiply by .
Step 5.4.2.2.1.4
Multiply .
Step 5.4.2.2.1.4.1
Multiply by .
Step 5.4.2.2.1.4.2
Multiply by .
Step 5.4.2.2.2
Subtract from .
Step 5.4.3
Evaluate .
Step 5.4.3.1
The determinant of a matrix can be found using the formula .
Step 5.4.3.2
Simplify the determinant.
Step 5.4.3.2.1
Simplify each term.
Step 5.4.3.2.1.1
Apply the distributive property.
Step 5.4.3.2.1.2
Multiply by .
Step 5.4.3.2.1.3
Multiply by .
Step 5.4.3.2.1.4
Multiply by .
Step 5.4.3.2.2
Step 5.4.4
Evaluate .
Step 5.4.4.1
The determinant of a matrix can be found using the formula .
Step 5.4.4.2
Simplify the determinant.
Step 5.4.4.2.1
Simplify each term.
Step 5.4.4.2.1.1
Multiply by .
Step 5.4.4.2.1.2
Multiply by .
Step 5.4.4.2.2
Subtract from .
Step 5.4.5
Simplify the determinant.
Step 5.4.5.1
Simplify each term.
Step 5.4.5.1.1
Apply the distributive property.
Step 5.4.5.1.2
Multiply by .
Step 5.4.5.1.3
Multiply by .
Step 5.4.5.1.4
Apply the distributive property.
Step 5.4.5.1.5
Multiply by .
Step 5.4.5.1.6
Multiply .
Step 5.4.5.1.6.1
Multiply by .
Step 5.4.5.1.6.2
Multiply by .
Step 5.4.5.1.7
Expand using the FOIL Method.
Step 5.4.5.1.7.1
Apply the distributive property.
Step 5.4.5.1.7.2
Apply the distributive property.
Step 5.4.5.1.7.3
Apply the distributive property.
Step 5.4.5.1.8
Simplify and combine like terms.
Step 5.4.5.1.8.1
Simplify each term.
Step 5.4.5.1.8.1.1
Multiply by .
Step 5.4.5.1.8.1.2
Multiply by .
Step 5.4.5.1.8.1.3
Rewrite using the commutative property of multiplication.
Step 5.4.5.1.8.1.4
Multiply by by adding the exponents.
Step 5.4.5.1.8.1.4.1
Move .
Step 5.4.5.1.8.1.4.2
Multiply by .
Step 5.4.5.1.8.1.5
Move to the left of .
Step 5.4.5.1.8.2
Step 5.4.5.1.9
Multiply by .
Step 5.4.5.2
Step 5.4.5.3
Subtract from .
Step 5.4.5.4
Subtract from .
Step 5.4.5.5
Reorder and .
Step 5.5
Evaluate .
Step 5.5.1
Choose the row or column with the most elements. If there are no elements choose any row or column. Multiply every element in row by its cofactor and add.
Step 5.5.1.1
Consider the corresponding sign chart.
Step 5.5.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 5.5.1.3
The minor for is the determinant with row and column deleted.
Step 5.5.1.4
Multiply element by its cofactor.
Step 5.5.1.5
The minor for is the determinant with row and column deleted.
Step 5.5.1.6
Multiply element by its cofactor.
Step 5.5.1.7
The minor for is the determinant with row and column deleted.
Step 5.5.1.8
Multiply element by its cofactor.
Step 5.5.1.9
Step 5.5.2
Evaluate .
Step 5.5.2.1
The determinant of a matrix can be found using the formula .
Step 5.5.2.2
Simplify the determinant.
Step 5.5.2.2.1
Simplify each term.
Step 5.5.2.2.1.1
Multiply by .
Step 5.5.2.2.1.2
Apply the distributive property.
Step 5.5.2.2.1.3
Multiply by .
Step 5.5.2.2.1.4
Multiply by .
Step 5.5.2.2.1.5
Apply the distributive property.
Step 5.5.2.2.1.6
Multiply by .
Step 5.5.2.2.1.7
Multiply by .
Step 5.5.2.2.2
Subtract from .
Step 5.5.2.2.3
Reorder and .
Step 5.5.3
Evaluate .
Step 5.5.3.1
The determinant of a matrix can be found using the formula .
Step 5.5.3.2
Simplify the determinant.
Step 5.5.3.2.1
Simplify each term.
Step 5.5.3.2.1.1
Multiply by .
Step 5.5.3.2.1.2
Apply the distributive property.
Step 5.5.3.2.1.3
Multiply by .
Step 5.5.3.2.1.4
Multiply by .
Step 5.5.3.2.2
Step 5.5.3.2.3
Reorder and .
Step 5.5.4
Evaluate .
Step 5.5.4.1
The determinant of a matrix can be found using the formula .
Step 5.5.4.2
Simplify the determinant.
Step 5.5.4.2.1
Simplify each term.
Step 5.5.4.2.1.1
Multiply by .
Step 5.5.4.2.1.2
Multiply by .
Step 5.5.4.2.2
Subtract from .
Step 5.5.5
Simplify the determinant.
Step 5.5.5.1
Simplify each term.
Step 5.5.5.1.1
Apply the distributive property.
Step 5.5.5.1.2
Multiply by .
Step 5.5.5.1.3
Multiply by .
Step 5.5.5.1.4
Apply the distributive property.
Step 5.5.5.1.5
Multiply by .
Step 5.5.5.1.6
Multiply .
Step 5.5.5.1.6.1
Multiply by .
Step 5.5.5.1.6.2
Multiply by .
Step 5.5.5.1.7
Expand using the FOIL Method.
Step 5.5.5.1.7.1
Apply the distributive property.
Step 5.5.5.1.7.2
Apply the distributive property.
Step 5.5.5.1.7.3
Apply the distributive property.
Step 5.5.5.1.8
Simplify and combine like terms.
Step 5.5.5.1.8.1
Simplify each term.
Step 5.5.5.1.8.1.1
Multiply by .
Step 5.5.5.1.8.1.2
Multiply by .
Step 5.5.5.1.8.1.3
Rewrite using the commutative property of multiplication.
Step 5.5.5.1.8.1.4
Multiply by by adding the exponents.
Step 5.5.5.1.8.1.4.1
Move .
Step 5.5.5.1.8.1.4.2
Multiply by .
Step 5.5.5.1.8.1.5
Move to the left of .
Step 5.5.5.1.8.2
Step 5.5.5.1.9
Multiply by .
Step 5.5.5.2
Step 5.5.5.3
Subtract from .
Step 5.5.5.4
Subtract from .
Step 5.5.5.5
Reorder and .
Step 5.6
Simplify the determinant.
Step 5.6.1
Simplify each term.
Step 5.6.1.1
Expand by multiplying each term in the first expression by each term in the second expression.
Step 5.6.1.2
Simplify each term.
Step 5.6.1.2.1
Multiply by .
Step 5.6.1.2.2
Multiply by .
Step 5.6.1.2.3
Multiply by .
Step 5.6.1.2.4
Multiply by .
Step 5.6.1.2.5
Rewrite using the commutative property of multiplication.
Step 5.6.1.2.6
Multiply by by adding the exponents.
Step 5.6.1.2.6.1
Move .
Step 5.6.1.2.6.2
Multiply by .
Step 5.6.1.2.6.2.1
Raise to the power of .
Step 5.6.1.2.6.2.2
Use the power rule to combine exponents.
Step 5.6.1.2.6.3
Step 5.6.1.2.7
Multiply by .
Step 5.6.1.2.8
Multiply by .
Step 5.6.1.2.9
Rewrite using the commutative property of multiplication.
Step 5.6.1.2.10
Multiply by by adding the exponents.
Step 5.6.1.2.10.1
Move .
Step 5.6.1.2.10.2
Multiply by .
Step 5.6.1.2.10.2.1
Raise to the power of .
Step 5.6.1.2.10.2.2
Use the power rule to combine exponents.
Step 5.6.1.2.10.3
Step 5.6.1.2.11
Multiply by .
Step 5.6.1.2.12
Rewrite using the commutative property of multiplication.
Step 5.6.1.2.13
Multiply by by adding the exponents.
Step 5.6.1.2.13.1
Move .
Step 5.6.1.2.13.2
Multiply by .
Step 5.6.1.2.14
Multiply by .
Step 5.6.1.2.15
Multiply by .
Step 5.6.1.3
Subtract from .
Step 5.6.1.4
Step 5.6.1.5
Subtract from .
Step 5.6.1.6
Apply the distributive property.
Step 5.6.1.7
Simplify.
Step 5.6.1.7.1
Multiply by .
Step 5.6.1.7.2
Multiply by .
Step 5.6.1.7.3
Multiply by .
Step 5.6.1.8
Apply the distributive property.
Step 5.6.1.9
Simplify.
Step 5.6.1.9.1
Multiply by .
Step 5.6.1.9.2
Multiply by .
Step 5.6.1.9.3
Multiply by .
Step 5.6.1.10
Apply the distributive property.
Step 5.6.1.11
Simplify.
Step 5.6.1.11.1
Multiply by .
Step 5.6.1.11.2
Multiply by .
Step 5.6.1.11.3
Multiply by .
Step 5.6.2
Combine the opposite terms in .
Step 5.6.2.1
Step 5.6.2.2
Step 5.6.3
Step 5.6.4
Step 5.6.5
Subtract from .
Step 5.6.6
Subtract from .
Step 5.6.7
Subtract from .
Step 5.6.8
Step 5.6.9
Step 5.6.10
Move .
Step 5.6.11
Move .
Step 5.6.12
Reorder and .