Linear Algebra Examples

Assign the matrix the name to simplify the descriptions throughout the problem.
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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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 .
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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.
Simplify element of the matrix.
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.
Move .
Use the power rule to combine exponents.
Add and to get .
Multiply by to get .
Simplify .
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Multiply by to get .
Multiply by to get .
Multiply by to get .
Expand by multiplying each term in the first expression by each term in the second expression.
Simplify terms.
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Remove unnecessary parentheses.
Simplify each term.
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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 .
Simplify by adding terms.
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Add and to get .
Subtract from to get .
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 to get .
Multiply by to get .
Simplify by multiplying through.
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Apply the distributive property.
Multiply.
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Multiply by to get .
Multiply by to get .
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 to get .
Apply the distributive property.
Simplify .
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Multiply by to get .
Multiply by to get .
Multiply by to get .
Apply the distributive property.
Move to the left of the expression .
Multiply by to get .
Multiply by to get .
Simplify the expression.
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Remove unnecessary parentheses.
Subtract from to get .
Add and to get .
Add and to get .
Simplify by adding and subtracting.
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Add and to get .
Subtract from to get .
Set the characteristic polynomial equal to to find the eigenvalues .
The roots of this equation could not be found algebraically, so the roots were determined numerically.
The eigenvector for is equal to the null space of the matrix minus the eigenvalue times the identity matrix.
Substitute the known values into the formula.
Simplify the matrix expression.
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Multiply by each element of the matrix.
Simplify each element of the matrix .
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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 .
Simplify each element of the matrix .
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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.
Simplify element of the matrix.
Simplify element of the matrix.
Simplify element of the matrix.
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 ).
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 ).
Use the result matrix to declare the final solutions to the system of equations.
This expression is the solution set for the system of equations.
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.
The null space of the set is the set of vectors created from the free variables of the system.
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