# 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.

Multiply by each element of the matrix.

Simplify each element of the matrix .

Rearrange .

Rearrange .

Rearrange .

Rearrange .

Add the corresponding elements of to each element of .

Simplify each element of the matrix .

Simplify .

Simplify .

Simplify .

Simplify .

These are both valid notations for the determinant of a matrix.

The determinant of a matrix can be found using the formula .

Simplify each term.

Multiply by .

Multiply by .

Multiply by .

Multiply by .

Rewrite as .

Since both terms are perfect squares, factor using the difference of squares formula, where and .

Set the characteristic polynomial equal to to find the eigenvalues .

If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .

Set the first factor equal to and solve.

Set the first factor equal to .

Subtract from both sides of the equation.

Set the next factor equal to and solve.

Set the next factor equal to .

Add to both sides of the equation.

The final solution is all the values that make true.

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.

Add the corresponding elements of to each element of .

Simplify each element of the matrix .

Simplify .

Simplify .

Simplify .

Simplify .

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.

Express the vector as a linear combination of column vector using the properties of vector column addition.

The null space of the set is the set of vectors created from the free variables of the system.

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.

Subtract the corresponding elements of from each element of .

Simplify each element of the matrix .

Simplify .

Simplify .

Simplify .

Simplify .

Perform the row operation on (row ) in order to convert some elements in the row to .

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 .

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.

Express the vector as a linear combination of column vector using the properties of vector column addition.

The null space of the set is the set of vectors created from the free variables of the system.

The eigenspace of is the union of the vector space for each eigenvalue.