Solve Using the Inverse
To solve an algebraic equation of the form where , multiply both sides of the equation by the multiplicative inverse of . Similarly, to solve a matrix equation of the form , multiply both sides of the equation by the inverse of .
Since matrix multiplication is not commutative, be sure to multiply both sides by in the same order. Use as the first factor in each product.
Solve Using Row Operations
Linear Equations Matrix
|System of Linear Equations||Corresponding Augmented Matrix|
The Gaussian elimination method is a way to solve systems of linear equations by using row operations. The steps are the same as when using the standard elimination method to solve systems of equations, but using a matrix helps to keep the numbers organized.
After translating the system into an augmented matrix, the goal is to reduce it so that all entries along the main diagonal of the coefficient matrix are 1s and all entries below the main diagonal are zeros. Then translate the matrix back to a system of equations, and use substitution to solve the simplified system.
Begin to use row operations to get 1s as entries along the main diagonal and 0s as entries below the main diagonal.
Start by getting zeros in the first column of rows 2 and 3.Multiply the first row by –2, and add the result to row 2.
Dependent and Inconsistent Systems
Independent, Dependent, and Inconsistent Systems
|Type of System||Reduced Form (After Row Operations)||Number of Solutions||Example|
|Consistent and independent||Ones along the main diagonal of the coefficient matrix||Exactly one solution|
|Inconsistent||Contains a row in which all entries are zero except the last entry||No solution|
|Dependent||Contains a row in which all entries are zero||Infinite number of solutions|