# 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.

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 .

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

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

Simplify the determinant.

Simplify each term.

Expand using the FOIL Method.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Simplify and combine like terms.

Simplify each term.

Multiply by .

Multiply by .

Multiply by .

Multiply by .

Multiply by .

Multiply by .

Subtract from .

Multiply by .

Subtract from .

Set the characteristic polynomial equal to to find the eigenvalues .

Use the quadratic formula to find the solutions.

Substitute the values , , and into the quadratic formula and solve for .

Simplify.

Simplify the numerator.

Raise to the power of .

Multiply by .

Multiply by .

Add and .

Multiply by .

Simplify the expression to solve for the portion of the .

Simplify the numerator.

Raise to the power of .

Multiply by .

Multiply by .

Add and .

Multiply by .

Change the to .

Simplify the expression to solve for the portion of the .

Simplify the numerator.

Raise to the power of .

Multiply by .

Multiply by .

Add and .

Multiply by .

Change the to .

The final answer is the combination of both solutions.

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.

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 .

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 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.

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 .

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 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.