# Linear Algebra Examples

Step 1

Write the matrix as a product of a lower triangular matrix and an upper triangular matrix.

Step 2

Step 2.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is and the second matrix is .

Step 2.2

Multiply each row in the first matrix by each column in the second matrix.

Step 2.3

Simplify each element of the matrix by multiplying out all the expressions.

Step 3

Step 3.1

Write as a linear system of equations.

Step 3.2

Solve the system of equations.

Step 3.2.1

Replace all occurrences of with in each equation.

Step 3.2.1.1

Replace all occurrences of in with .

Step 3.2.1.2

Simplify the left side.

Step 3.2.1.2.1

Multiply by .

Step 3.2.1.3

Replace all occurrences of in with .

Step 3.2.1.4

Simplify the left side.

Step 3.2.1.4.1

Multiply by .

Step 3.2.2

Replace all occurrences of with in each equation.

Step 3.2.2.1

Replace all occurrences of in with .

Step 3.2.2.2

Simplify the left side.

Step 3.2.2.2.1

Multiply by .

Step 3.2.2.3

Replace all occurrences of in with .

Step 3.2.2.4

Simplify the left side.

Step 3.2.2.4.1

Multiply by .

Step 3.2.3

Replace all occurrences of with in each equation.

Step 3.2.3.1

Replace all occurrences of in with .

Step 3.2.3.2

Simplify the left side.

Step 3.2.3.2.1

Multiply by .

Step 3.2.3.3

Replace all occurrences of in with .

Step 3.2.3.4

Simplify the left side.

Step 3.2.3.4.1

Multiply by .

Step 3.2.4

Replace all occurrences of with in each equation.

Step 3.2.4.1

Replace all occurrences of in with .

Step 3.2.4.2

Simplify the left side.

Step 3.2.4.2.1

Multiply by .

Step 3.2.4.3

Replace all occurrences of in with .

Step 3.2.4.4

Simplify the left side.

Step 3.2.4.4.1

Remove parentheses.

Step 3.2.5

Replace all occurrences of with in each equation.

Step 3.2.5.1

Move all terms not containing to the right side of the equation.

Step 3.2.5.1.1

Subtract from both sides of the equation.

Step 3.2.5.1.2

Subtract from .

Step 3.2.5.2

Replace all occurrences of in with .

Step 3.2.5.3

Simplify the left side.

Step 3.2.5.3.1

Multiply by .

Step 3.2.5.4

Replace all occurrences of in with .

Step 3.2.5.5

Simplify the left side.

Step 3.2.5.5.1

Remove parentheses.

Step 3.2.6

Replace all occurrences of with in each equation.

Step 3.2.6.1

Move all terms not containing to the right side of the equation.

Step 3.2.6.1.1

Subtract from both sides of the equation.

Step 3.2.6.1.2

Subtract from .

Step 3.2.6.2

Replace all occurrences of in with .

Step 3.2.6.3

Simplify the left side.

Step 3.2.6.3.1

Move to the left of .

Step 3.2.7

Replace all occurrences of with in each equation.

Step 3.2.7.1

Move all terms not containing to the right side of the equation.

Step 3.2.7.1.1

Subtract from both sides of the equation.

Step 3.2.7.1.2

Subtract from .

Step 3.2.7.2

Divide each term in by and simplify.

Step 3.2.7.2.1

Divide each term in by .

Step 3.2.7.2.2

Simplify the left side.

Step 3.2.7.2.2.1

Cancel the common factor of .

Step 3.2.7.2.2.1.1

Cancel the common factor.

Step 3.2.7.2.2.1.2

Divide by .

Step 3.2.7.2.3

Simplify the right side.

Step 3.2.7.2.3.1

Dividing two negative values results in a positive value.

Step 3.2.7.3

Replace all occurrences of in with .

Step 3.2.7.4

Simplify the left side.

Step 3.2.7.4.1

Move to the left of .

Step 3.2.8

Replace all occurrences of with in each equation.

Step 3.2.8.1

Replace all occurrences of in with .

Step 3.2.8.2

Simplify the left side.

Step 3.2.8.2.1

Simplify .

Step 3.2.8.2.1.1

Simplify each term.

Step 3.2.8.2.1.1.1

Cancel the common factor of .

Step 3.2.8.2.1.1.1.1

Factor out of .

Step 3.2.8.2.1.1.1.2

Cancel the common factor.

Step 3.2.8.2.1.1.1.3

Rewrite the expression.

Step 3.2.8.2.1.1.2

Multiply by .

Step 3.2.8.2.1.2

Subtract from .

Step 3.2.9

Move all terms not containing to the right side of the equation.

Step 3.2.9.1

Add to both sides of the equation.

Step 3.2.9.2

Add and .

Step 3.2.10

Solve the system of equations.

Step 3.2.11

List all of the solutions.

Step 4

Substitute in the solved values.