Algebra Examples

Step 1
Assign the matrix the name to simplify the descriptions throughout the problem.
Step 2
Set up the formula to find the characteristic equation .
Step 3
Substitute the known values in the formula.
Step 4
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 in the matrix.
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Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Add the corresponding elements.
Simplify each element of the matrix .
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Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Step 5
Find the determinant of .
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Set up the determinant by breaking it into smaller components.
Find the determinant of .
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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 .
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
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Move .
Multiply by .
Multiply by .
Multiply by .
Subtract from .
Multiply by .
Subtract from .
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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Multiply by .
Multiply by .
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
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Move .
Multiply by .
Multiply by .
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 .
Simplify by adding terms.
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Add and .
Add and .
Find the determinant of .
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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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Add and .
Apply the distributive property.
Multiply.
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Multiply by .
Multiply by .
Find the determinant of .
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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 by .
Multiply .
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Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Move to the left of .
Simplify by multiplying through.
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Add and .
Apply the distributive property.
Multiply.
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Multiply by .
Multiply by .
Subtract from .
Subtract from .
Subtract from .
Subtract from .
Step 6
Reorder the polynomial.
Step 7
Set the characteristic polynomial equal to to find the eigenvalues .
Step 8
Find the roots of by solving for .
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Factor the left side of the equation.
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Factor using the rational roots test.
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If a polynomial function has integer coefficients, then every rational zero will have the form where is a factor of the constant and is a factor of the leading coefficient.
Find every combination of . These are the possible roots of the polynomial function.
Substitute and simplify the expression. In this case, the expression is equal to so is a root of the polynomial.
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Substitute into the polynomial.
Raise to the power of .
Multiply by .
Raise to the power of .
Add and .
Multiply by .
Subtract from .
Add and .
Since is a known root, divide the polynomial by to find the quotient polynomial. This polynomial can then be used to find the remaining roots.
Divide by .
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Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of .
+-+++
Divide the highest order term in the dividend by the highest order term in divisor .
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+-+++
Multiply the new quotient term by the divisor.
-
+-+++
--
The expression needs to be subtracted from the dividend, so change all the signs in
-
+-+++
++
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
-
+-+++
++
+
Pull the next terms from the original dividend down into the current dividend.
-
+-+++
++
++
Divide the highest order term in the dividend by the highest order term in divisor .
-+
+-+++
++
++
Multiply the new quotient term by the divisor.
-+
+-+++
++
++
++
The expression needs to be subtracted from the dividend, so change all the signs in
-+
+-+++
++
++
--
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
-+
+-+++
++
++
--
+
Pull the next terms from the original dividend down into the current dividend.
-+
+-+++
++
++
--
++
Divide the highest order term in the dividend by the highest order term in divisor .
-++
+-+++
++
++
--
++
Multiply the new quotient term by the divisor.
-++
+-+++
++
++
--
++
++
The expression needs to be subtracted from the dividend, so change all the signs in
-++
+-+++
++
++
--
++
--
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
-++
+-+++
++
++
--
++
--
Since the remander is , the final answer is the quotient.
Write as a set of factors.
Factor by grouping.
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Factor by grouping.
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For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
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Factor out of .
Rewrite as plus
Apply the distributive property.
Factor out the greatest common factor from each group.
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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.
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Set equal to and solve for .
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Set equal to .
Subtract from both sides of the equation.
Set equal to and solve for .
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Set equal to .
Add to both sides of the equation.
The final solution is all the values that make true.
Step 9
The eigenvector for is equal to the null space of the matrix minus the eigenvalue times the identity matrix.
Step 10
Substitute the known values into the formula.
Step 11
Simplify the matrix expression.
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Add the corresponding elements.
Simplify each element of the matrix .
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Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Step 12
Find the reduced row echelon form of the matrix.
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Perform the row operation on (row ) in order to convert some elements in the row to .
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Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Perform the row operation on (row ) in order to convert some elements in the row to .
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Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Perform the row operation on (row ) in order to convert some elements in the row to .
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Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Step 13
Use the result matrix to declare the final solutions to the system of equations.
Step 14
This expression is the solution set for the system of equations.
Step 15
Decompose a solution vector by re-arranging each equation represented in the row-reduced form of the augmented matrix by solving for the dependent variable in each row yields the vector equality.
Step 16
Express the vector as a linear combination of column vector using the properties of vector column addition.
Step 17
The null space of the set is the set of vectors created from the free variables of the system.
Step 18
The eigenvector for is equal to the null space of the matrix minus the eigenvalue times the identity matrix.
Step 19
Substitute the known values into the formula.
Step 20
Simplify the matrix expression.
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Multiply by each element of the matrix.
Simplify each element in the matrix.
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Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Rearrange .
Add the corresponding elements.
Simplify each element of the matrix .
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Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Simplify .
Step 21
Find the reduced row echelon form of the matrix.
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Perform the row operation on (row ) in order to convert some elements in the row to .
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Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Perform the row operation on (row ) in order to convert some elements in the row to .
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Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Perform the row operation on (row ) in order to convert some elements in the row to .
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Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Perform the row operation on (row ) in order to convert some elements in the row to .
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Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Perform the row operation on (row ) in order to convert some elements in the row to .
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Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Perform the row operation on (row ) in order to convert some elements in the row to .
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Replace (row ) with the row operation in order to convert some elements in the row to the desired value .
Replace (row ) with the actual values of the elements for the row operation .
Simplify (row ).
Step 22
Use the result matrix to declare the final solutions to the system of equations.
Step 23
This expression is the solution set for the system of equations.
Step 24
Decompose a solution vector by re-arranging each equation represented in the row-reduced form of the augmented matrix by solving for the dependent variable in each row yields the vector equality.
Step 25
Express the vector as a linear combination of column vector using the properties of vector column addition.
Step 26
The null space of the set is the set of vectors created from the free variables of the system.
Step 27
The eigenspace of is the union of the vector space for each eigenvalue.
Step 28
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