# 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 .

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 .

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.

Set up the determinant by breaking it into smaller components.

The determinant of is .

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

Simplify the determinant.

Simplify terms.

Simplify each term.

Expand using the FOIL Method.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Remove parentheses.

Simplify and combine like terms.

Simplify each term.

Move .

Use the power rule to combine exponents.

Add and to get .

Simplify .

Multiply by to get .

Multiply by to get .

Multiply by to get .

Multiply by to get .

Multiply by to get .

Subtract from to get .

Multiply by to get .

Subtract from to get .

Expand by multiplying each term in the first expression by each term in the second expression.

Simplify terms.

Remove unnecessary parentheses.

Simplify each term.

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.

Subtract from to get .

Add and to get .

The determinant of is .

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

Simplify the determinant.

Simplify each term.

Apply the distributive property.

Multiply by to get .

Multiply by to get .

Multiply by to get .

Simplify by multiplying through.

Add and to get .

Apply the distributive property.

Multiply.

Multiply by to get .

Multiply by to get .

The determinant of is .

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

Simplify the determinant.

Simplify each term.

Multiply by to get .

Apply the distributive property.

Simplify .

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 by multiplying through.

Remove unnecessary parentheses.

Add and to get .

Apply the distributive property.

Multiply.

Multiply by to get .

Multiply by to get .

Subtract from to get .

Subtract from to get .

Simplify by subtracting numbers.

Subtract from to get .

Subtract from to get .

Set the characteristic polynomial equal to to find the eigenvalues .

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 the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is , which means it is a root.

Simplify the expression. In this case, the expression is equal to so is a root of the polynomial.

Simplify each term.

Use the power rule to combine exponents.

Add and to get .

Raise to the power of to get .

Raise to the power of to get .

Multiply by to get .

Simplify by adding and subtracting.

Add and to get .

Subtract from to get .

Add and to get .

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.

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by .

Place the numbers representing the divisor and the dividend into a division-like configuration.

The first number in the dividend is put into the first position of the result area (below the horizontal line).

Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

Factor by grouping.

For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .

Factor out of .

Rewrite as plus

Apply the distributive property.

Remove parentheses.

Factor out the greatest common factor from each group.

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, .

Rewrite as .

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 .

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 .

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.

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 ).

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 ).

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 ).

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.