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.
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Multiply by each element of the matrix.
Simplify each element of the matrix .
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Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Add the corresponding elements of to each element of .
Simplify each element of the matrix .
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Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
The determinant of is .
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Set up the determinant by breaking it into smaller components.
The determinant of is .
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Set up the determinant by breaking it into smaller components.
The determinant of is .
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The determinant of a matrix can be found using the formula .
Simplify the determinant.
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Simplify terms.
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Simplify each term.
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Expand using the FOIL Method.
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Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
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Simplify each term.
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Multiply by .
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Add and .
Multiply by .
Combine the opposite terms in .
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Add and .
Add and .
Expand using the FOIL Method.
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Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
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Simplify each term.
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Multiply by by adding the exponents.
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Move .
Multiply by .
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Raise to the power of .
Use the power rule to combine exponents.
Add and .
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Add and .
The determinant of is .
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The determinant of a matrix can be found using the formula .
Simplify the determinant.
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Simplify each term.
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Apply the distributive property.
Multiply by .
Multiply by .
Multiply by .
Simplify by multiplying through.
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Subtract from .
Apply the distributive property.
Multiply.
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Multiply by .
Multiply by .
The determinant of is .
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The determinant of a matrix can be found using the formula .
Simplify the determinant.
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Simplify each term.
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Multiply by .
Apply the distributive property.
Multiply .
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Multiply by .
Multiply by .
Multiply by .
Apply the distributive property.
Move to the left of .
Multiply by .
Simplify by multiplying through.
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Add and .
Apply the distributive property.
Multiply.
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Multiply by .
Multiply by .
Add and .
Subtract from .
Subtract from .
Simplify the determinant.
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Expand by multiplying each term in the first expression by each term in the second expression.
Simplify terms.
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Simplify each term.
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Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
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Multiply by .
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Raise to the power of .
Use the power rule to combine exponents.
Add and .
Multiply by .
Multiply by .
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
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Multiply by .
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Raise to the power of .
Use the power rule to combine exponents.
Add and .
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Simplify by adding terms.
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Add and .
Subtract from .
Add and .
The determinant of is .
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Set up the determinant by breaking it into smaller components.
The determinant of is .
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The determinant of a matrix can be found using the formula .
Simplify the determinant.
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Simplify each term.
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Expand using the FOIL Method.
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Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
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Simplify each term.
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Multiply by .
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Add and .
Multiply by .
Combine the opposite terms in .
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Add and .
Add and .
The determinant of is .
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The determinant of a matrix can be found using the formula .
Simplify the determinant.
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Simplify each term.
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Apply the distributive property.
Multiply by .
Multiply by .
Multiply by .
Simplify by adding terms.
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Combine the opposite terms in .
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Add and .
Add and .
Multiply by .
The determinant of is .
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The determinant of a matrix can be found using the formula .
Simplify the determinant.
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Simplify each term.
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Multiply by .
Apply the distributive property.
Multiply .
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Multiply by .
Multiply by .
Multiply by .
Apply the distributive property.
Move to the left of .
Multiply by .
Simplify by adding terms.
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Combine the opposite terms in .
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Subtract from .
Add and .
Multiply by .
Subtract from .
Add and .
Simplify the determinant.
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Apply the distributive property.
Multiply by .
The determinant of is .
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Set up the determinant by breaking it into smaller components.
The determinant of is .
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The determinant of a matrix can be found using the formula .
Simplify the determinant.
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Simplify each term.
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Apply the distributive property.
Multiply by .
Multiply by .
Multiply by .
Subtract from .
The determinant of is .
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The determinant of a matrix can be found using the formula .
Simplify the determinant.
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Apply the distributive property.
Multiply .
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Multiply by .
Multiply by .
Multiply by .
Simplify each term.
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Apply the distributive property.
Multiply by .
Multiply by .
Multiply by .
Simplify terms.
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Combine the opposite terms in .
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Add and .
Add and .
Apply the distributive property.
Simplify the expression.
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Rewrite using the commutative property of multiplication.
Multiply by .
Multiply by by adding the exponents.
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Move .
Multiply by .
The determinant of is .
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The determinant of a matrix can be found using the formula .
Simplify the determinant.
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Simplify each term.
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Multiply by .
Multiply by .
Simplify the expression.
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Add and .
Multiply by .
Subtract from .
Add and .
Simplify the determinant.
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Apply the distributive property.
Simplify.
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Multiply by .
Multiply by .
Multiply by .
The determinant of is .
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Set up the determinant by breaking it into smaller components.
The determinant of is .
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The determinant of a matrix can be found using the formula .
Simplify the determinant.
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Simplify each term.
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Multiply by .
Apply the distributive property.
Multiply .
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Multiply by .
Multiply by .
Multiply by .
Apply the distributive property.
Move to the left of .
Multiply by .
Add and .
The determinant of is .
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The determinant of a matrix can be found using the formula .
Simplify the determinant.
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Apply the distributive property.
Multiply .
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Multiply by .
Multiply by .
Multiply by .
Simplify each term.
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Multiply by .
Apply the distributive property.
Multiply .
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Multiply by .
Multiply by .
Multiply by .
Apply the distributive property.
Move to the left of .
Multiply by .
Simplify terms.
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Combine the opposite terms in .
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Subtract from .
Add and .
Apply the distributive property.
Simplify the expression.
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Rewrite using the commutative property of multiplication.
Multiply by .
Multiply by by adding the exponents.
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Move .
Multiply by .
The determinant of is .
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The determinant of a matrix can be found using the formula .
Simplify the determinant.
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Simplify each term.
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Multiply by .
Multiply by .
Simplify the expression.
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Add and .
Multiply by .
Add and .
Subtract from .
Simplify the determinant.
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Apply the distributive property.
Simplify.
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Multiply by .
Multiply by .
Multiply by .
Combine the opposite terms in .
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Add and .
Add and .
Add and .
Add and .
Subtract from .
Subtract from .
Simplify by adding numbers.
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Add and .
Add and .
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