# Linear Algebra Examples

Solve Using a Matrix with Cramer's Rule
,
Step 1
Represent the system of equations in matrix format.
Step 2
Find the determinant of the coefficient matrix .
Step 2.1
Write in determinant notation.
Step 2.2
The determinant of a matrix can be found using the formula .
Step 2.3
Simplify the determinant.
Step 2.3.1
Simplify each term.
Step 2.3.1.1
Multiply by .
Step 2.3.1.2
Multiply by .
Step 2.3.2
Step 3
Since the determinant is not , the system can be solved using Cramer's Rule.
Step 4
Find the value of by Cramer's Rule, which states that .
Step 4.1
Replace column of the coefficient matrix that corresponds to the -coefficients of the system with .
Step 4.2
Find the determinant.
Step 4.2.1
The determinant of a matrix can be found using the formula .
Step 4.2.2
Simplify the determinant.
Step 4.2.2.1
Simplify each term.
Step 4.2.2.1.1
Multiply by .
Step 4.2.2.1.2
Multiply by .
Step 4.2.2.2
Step 4.3
Use the formula to solve for .
Step 4.4
Substitute for and for in the formula.
Step 4.5
Divide by .
Step 5
Find the value of by Cramer's Rule, which states that .
Step 5.1
Replace column of the coefficient matrix that corresponds to the -coefficients of the system with .
Step 5.2
Find the determinant.
Step 5.2.1
The determinant of a matrix can be found using the formula .
Step 5.2.2
Simplify the determinant.
Step 5.2.2.1
Simplify each term.
Step 5.2.2.1.1
Multiply by .
Step 5.2.2.1.2
Multiply by .
Step 5.2.2.2