Row Echelon Form Calculator

Row Echelon Form (REF) Calculator

Enter your 3×3 matrix values below:

What is Row Echelon Form?

In linear algebra, a matrix is in Row Echelon Form (REF) if it satisfies three specific conditions resulting from Gaussian elimination. This form is essential for solving systems of linear equations, finding matrix rank, and understanding the vector space spanned by the rows.

• Leading Entries: The first non-zero number in a row (called the pivot) must be to the right of the leading entry of the row above it.
• Zero Rows: Any rows consisting entirely of zeros must be located at the bottom of the matrix.
• Staircase Pattern: All entries below a leading entry must be zeros.

Difference Between REF and RREF

While Row Echelon Form requires zeros only below the pivots, Reduced Row Echelon Form (RREF) goes a step further. In RREF, every leading entry must be 1, and it must be the only non-zero entry in its column (zeros above and below the pivot).

How Gaussian Elimination Works

To transform a matrix into REF, we use three primary row operations:

1. Swapping: Interchange two rows to position a non-zero element in a pivot position.
2. Scaling: Multiply a row by a non-zero constant.
3. Addition: Add or subtract a multiple of one row to another row to create zeros in a specific column.

Example Calculation:
Suppose you have the matrix:
[2, 4, -2]
[4, 9, -3]
[-2, -3, 7]

1. Use the first row to eliminate the '4' in row 2: R2 = R2 – 2*R1.
2. Use the first row to eliminate the '-2' in row 3: R3 = R3 + R1.
3. Use the new row 2 to eliminate the entry below its pivot in row 3.

Why Use Row Echelon Form?

REF is the first major step in solving linear systems. Once a matrix is in this form, you can use back-substitution to find the values of your variables. It also reveals the Rank of the matrix, which tells you the number of linearly independent rows or columns present in the system.

Matrix Rank: