91Ó°ÊÓ

A row-reduced echelon form, denoted as RREF, is a form of a matrix that is obtained through a process called Gaussian elimination or row reduction. By applying a series of row operations, a matrix can be transformed into a simpler form that has distinct properties, which make it easier to analyze and solve linear systems. The objective of finding the RREF of a matrix is to identify the unique solution, no solution, or infinitely many solutions for the corresponding system of linear equations.

A matrix is in row-reduced echelon form if it satisfies the following properties: 1. All zero rows, if any, are at the bottom of the matrix. 2. The first nonzero number from the left (also called the leading entry or pivot) in a nonzero row is always strictly to the right of the pivot in the row above. 3. All entries above and below a pivot are zero. 4. The pivot in each nonzero row is 1.

We will now provide an example of converting a matrix to its row-reduced echelon form. Given matrix A: \[A = \begin{bmatrix} 1 & 2 & -1 & -4 \\ 2 & 3 &-1 & -11 \\ -2 & 0 &-3 & 22 \end{bmatrix}\] We will perform row operations to obtain its RREF: 1. Subtract 2 times row 1 from row 2: \[\begin{bmatrix} 1 & 2 & -1 & -4 \\ 0 & -1 & 1 & 3 \\ -2 & 0 & -3 & 22 \end{bmatrix}\] 2. Subtract -2 times row 1 from row 3: \[\begin{bmatrix} 1 & 2 & -1 & -4 \\ 0 & -1 & 1 & 3 \\ 0 & 4 & -5 & 14 \end{bmatrix}\] 3. Multiply row 2 by -1: \[\begin{bmatrix} 1 & 2 & -1 & -4 \\ 0 & 1 & -1 & -3 \\ 0 & 4 & -5 & 14 \end{bmatrix}\] 4. Subtract 2 times row 2 from row 1 and 4 times row 2 from row 3: \[\begin{bmatrix} 1 & 0 & 1 & 2 \\ 0 & 1 & -1 & -3 \\ 0 & 0 & 1 & 2 \end{bmatrix}\] 5. Subtract row 3 from row 1, and add row 3 to row 2: \[\begin{bmatrix} 1 & 0 & 0 &4 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 2 \end{bmatrix}\] The matrix is now in row-reduced echelon form.