UNIT 8 • MATRICES & LINEAR ALGEBRA

Rank of a Matrix
Row Echelon Form (RREF)

The rank $\rho(A)$ of a matrix is the number of non-zero rows left after you reduce it to echelon form. It counts how many rows (or columns) are truly independent.

HOW RANK IS FOUNDANY m × n MATRIX

\rho(A) = \text{number of non-zero rows in the echelon form of } A \ \le\ \min(m, n)

1. PivotTake the first column with a non-zero entry; swap a row up if needed and make the pivot 1.

2. Clear the ColumnUse

R_i \to R_i - k R_j

to make every other entry in that column 0.

3. CountRepeat for the next column. The rank is the number of pivots (non-zero rows).

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Find the rank of $A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 1 & 1 \end{bmatrix}$ .

STEP 01 • CLEAR COLUMN 1

The pivot is the 1 in row 1. Subtract multiples of row 1 from the rows below:

R_2 \to R_2 - 2R_1,\quad R_3 \to R_3 - R_1:\quad \begin{bmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & -1 & -2 \end{bmatrix}

STEP 02 • PIVOT IN COLUMN 2

Row 2 now has a 0 in column 2, so swap it with row 3, then make the pivot 1:

R_2 \leftrightarrow R_3,\quad R_2 \to R_2 \div (-1):\quad \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{bmatrix}

STEP 03 • CLEAR ABOVE THE PIVOT

R_1 \to R_1 - 2R_2:\quad \begin{bmatrix} 1 & 0 & -1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{bmatrix}

Column 3 has no pivot left (only zeros below row 2), so the reduction is finished. This is the RREF.

FINAL CONCLUSION

ρ(A) = 2

\text{2 non-zero rows} \implies \rho(A) = 2

Row 2 was just 2 × row 1, so only two rows are independent. That also means |A| = 0.

LABORATORY

Interactive Rank Calculator

LABORATORY

Rank & RREF Engine

Rows:Columns:

A =

Common Questions

What is the rank of a matrix?

The number of linearly independent rows (which always equals the number of independent columns). In practice: row-reduce the matrix and count the rows that are not all zeros.

What is the difference between echelon form and reduced row echelon form (RREF)?

In echelon form, each leading entry is to the right of the one above and everything below it is 0. RREF also makes every leading entry 1 and clears the entries above it. Both give the same rank.

What is the rank of a 3×3 matrix with determinant 0?

Less than 3. A square matrix has full rank exactly when its determinant is not zero.

How is rank used to check a system of equations?

Compare the rank of the coefficient matrix

A

with the rank of the augmented matrix

[A \mid b]

. If they are equal the system is consistent; if they also equal the number of unknowns, the solution is unique.

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Unit 8•37 of 47

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Evaluating numerical algorithm

UNIT 8 • MATRICES & LINEAR ALGEBRA

Rank of a Matrix
Row Echelon Form (RREF)

The rank $\rho(A)$ of a matrix is the number of non-zero rows left after you reduce it to echelon form. It counts how many rows (or columns) are truly independent.

HOW RANK IS FOUNDANY m × n MATRIX

\rho(A) = \text{number of non-zero rows in the echelon form of } A \ \le\ \min(m, n)

1. PivotTake the first column with a non-zero entry; swap a row up if needed and make the pivot 1.

2. Clear the ColumnUse

R_i \to R_i - k R_j

to make every other entry in that column 0.

3. CountRepeat for the next column. The rank is the number of pivots (non-zero rows).

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Find the rank of $A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 1 & 1 \end{bmatrix}$ .

STEP 01 • CLEAR COLUMN 1

The pivot is the 1 in row 1. Subtract multiples of row 1 from the rows below:

R_2 \to R_2 - 2R_1,\quad R_3 \to R_3 - R_1:\quad \begin{bmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & -1 & -2 \end{bmatrix}

STEP 02 • PIVOT IN COLUMN 2

Row 2 now has a 0 in column 2, so swap it with row 3, then make the pivot 1:

R_2 \leftrightarrow R_3,\quad R_2 \to R_2 \div (-1):\quad \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{bmatrix}

STEP 03 • CLEAR ABOVE THE PIVOT

R_1 \to R_1 - 2R_2:\quad \begin{bmatrix} 1 & 0 & -1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{bmatrix}

Column 3 has no pivot left (only zeros below row 2), so the reduction is finished. This is the RREF.

FINAL CONCLUSION

ρ(A) = 2

\text{2 non-zero rows} \implies \rho(A) = 2

Row 2 was just 2 × row 1, so only two rows are independent. That also means |A| = 0.

LABORATORY

Interactive Rank Calculator

LABORATORY

Rank & RREF Engine

Rows:Columns:

A =

Common Questions

What is the rank of a matrix?

The number of linearly independent rows (which always equals the number of independent columns). In practice: row-reduce the matrix and count the rows that are not all zeros.

What is the difference between echelon form and reduced row echelon form (RREF)?

In echelon form, each leading entry is to the right of the one above and everything below it is 0. RREF also makes every leading entry 1 and clears the entries above it. Both give the same rank.

What is the rank of a 3×3 matrix with determinant 0?

Less than 3. A square matrix has full rank exactly when its determinant is not zero.

How is rank used to check a system of equations?

Compare the rank of the coefficient matrix

A

with the rank of the augmented matrix

[A \mid b]

. If they are equal the system is consistent; if they also equal the number of unknowns, the solution is unique.

PreviousInverse of a Matrix

Unit 8•37 of 47

NextMatrix Calculator

Rank of a Matrix Row Echelon Form (RREF)

Step-by-Step Worked Example

ρ(A) = 2

Interactive Rank Calculator

Rank & RREF Engine

Common Questions

NETZ Engine

Rank of a Matrix Row Echelon Form (RREF)

Step-by-Step Worked Example

ρ(A) = 2

Interactive Rank Calculator

Rank & RREF Engine

Common Questions

Rank of a Matrix
Row Echelon Form (RREF)

Rank of a Matrix
Row Echelon Form (RREF)