Step 2c: Use row replacement so all entries below this 1 are 0.Step 2b: Scale the 2nd row so that its first nonzero entry is equal to 1.Step 2a: Swap the 2nd row with a lower one so that the leftmost nonzero entry is in the 2nd row.Step 1c: Use row replacement so all entries below this 1 are 0.Step 1b: Scale the 1st row so that its first nonzero entry is equal to 1.Step 1a: Swap the 1st row with a lower one so a leftmost nonzero entry is in the 1st row (if necessary).We will not prove uniqueness, but maybe you can! Algorithm (Row Reduction) This assumes, of course, that you only do the three legal row operations, and you don’t make any arithmetic errors. The uniqueness statement is interesting-it means that, no matter how you row reduce, you always get the same matrix in reduced row echelon form.

We will give an algorithm, called row reduction or Gaussian elimination, which demonstrates that every matrix is row equivalent to at least one matrix in reduced row echelon form. Subsection 1.2.3 The Row Reduction Algorithm TheoremĮvery matrix is row equivalent to one and only one matrix in reduced row echelon form. This is one possible explanation for the terminology “pivot”. We used the pivot position in the matrix in order to make the blue line pivot like this. We can think of the blue line as rotating, or pivoting, around the solution ( 1,1 ). What has happened geometrically is that the original blue line has been replaced with the new blue line y = 1. Scaling: we can multiply both sides of an equation by a nonzero number.There are three valid operations we can perform on our system of equations: In other words, we will combine the equations in various ways to try to eliminate as many variables as possible from each equation. We will solve systems of linear equations algebraically using the elimination method. Subsection 1.2.1 The Elimination Method ¶ In this section, we will present an algorithm for “solving” a system of linear equations. Vocabulary words: row operation, row equivalence, matrix, augmented matrix, pivot, (reduced) row echelon form.Learn which row reduced matrices come from inconsistent linear systems.Understand when a matrix is in (reduced) row echelon form.Learn how the elimination method corresponds to performing row operations on an augmented matrix.Learn to replace a system of linear equations by an augmented matrix.Hints and Solutions to Selected Exercises.The adjugate matrix is noted as Adj(M).3 Linear Transformations and Matrix Algebra This is sometimes referred to as the adjoint matrix. The final result of this step is called the adjugate matrix of the original.They are indicators of keeping (+) or reversing (-) whatever sign the number originally had. Note that the (+) or (-) signs in the checkerboard diagram do not suggest that the final term should be positive or negative.

Continue on with the rest of the matrix in this fashion. The third element keeps its original sign.

When assigning signs, the first element of the first row keeps its original sign.

You must then reverse the sign of alternating terms of this new matrix, following the “checkerboard” pattern shown. Thus, the determinant that you calculated from item (1,1) of the original matrix goes in position (1,1). Place the results of the previous step into a new matrix of cofactors by aligning each minor matrix determinant with the corresponding position in the original matrix.

Augmented matrix 3x3