Matrix Rank

Minors and Rank

Consider a matrix $A_{m \times n}$ . Choose arbitrary $k$ rows and $k$ columns in it ( $1 \leq k \leq min (m, n)$ ). The entries at the intersections form a square matrix of order $k$ .

Definition. A minor of order $k$ of the matrix $A$ is the determinant formed from the entries at the intersections of the chosen $k$ rows and $k$ columns.

For example, for a $3 \times 3$ matrix, the order-2 minors are obtained by deleting one row and one column:

a_{11} a_{21} a_{12} a_{22}, a_{12} a_{22} a_{13} a_{23}, \dots

Definition. The rank of a matrix $A$ is the highest order of a nonzero minor. Notation: $r (A)$ , $rank A$ , $r$ .

Remarks

$r (A) = 0 ⟺ A = O$ (the zero matrix).
$r (A) = k$ if and only if two conditions hold:
- (a) there exists a nonzero minor of order $k$ ( $\exists M_{k} \neq = 0$ );
- (b) all minors of order $k + 1$ are zero (or do not exist).

The Rank of a Row-Echelon Matrix

Theorem. The rank of a row-echelon matrix equals the number of its nonzero rows (the number of "steps").

Proof. Suppose the row-echelon matrix has $k$ nonzero rows with leading entries $a_{i_{1} j_{1}}, a_{i_{2} j_{2}}, \dots, a_{i_{k} j_{k}}$ . The minor formed from these rows and columns $j_{1}, \dots, j_{k}$ is triangular, and its determinant

a_{i_{1} j_{1}} 0 ⋮ 0 * a_{i_{2} j_{2}} 0 \dots \dots ⋱ \dots * * ⋮ a_{i_{k} j_{k}} = a_{i_{1} j_{1}} a_{i_{2} j_{2}} \dots a_{i_{k} j_{k}} \neq = 0.

Any minor of higher order contains a zero row, hence equals zero. So $r (A) = k$ .

Invariance of Rank Under Elementary Transformations

Theorem. Under elementary row (and column) transformations, the rank of a matrix does not change.

Recall the elementary row transformations: (1) multiplying a row by a number $λ \neq = 0$ ; (2) swapping two rows; (3) adding to one row another, multiplied by a number.

Proof. For a transformation of type 3; for types 1 and 2 the claim follows from the properties of the determinant. Let $B$ be obtained from $A$ by adding to row $i$ row $j$ multiplied by $λ$ , and let $r (A) = k$ . Take an arbitrary minor $D$ of $B$ of order higher than $k$ and show that $D = 0$ . There are three possible cases:

$D$ does not contain the modified row $i$ — then $D$ coincides with a minor of $A$ , so $D = 0$ .
$D$ contains both row $i$ and row $j$ — by linearity of the determinant, the added $λ \cdot (row j)$ produces a minor with two proportional rows, so $D$ equals the corresponding minor of $A$ , i.e. $D = 0$ .
$D$ contains row $i$ but not row $j$ — then $D = D_{1} + λ D_{2}$ , where $D_{1}$ is a minor of $A$ , and $D_{2}$ is also a minor of $A$ (of order higher than $k$ ); both are zero, so $D = 0$ .

So $r (B) \leq r (A)$ . Since the transformation is reversible (one can go back from $B$ to $A$ ), similarly $r (A) \leq r (B)$ . Hence $r (A) = r (B)$ .

Remark. The same holds for column transformations: rank is preserved under elementary transformations of both rows and columns.

Practical Computation of Rank

The two preceding theorems give a convenient method: reduce the matrix by elementary transformations to row-echelon form and count the number of nonzero rows.

Example. Let's find the rank of

A = 121241361 .

The transformations $R_{2} \to R_{2} - 2 R_{1}$ and $R_{3} \to R_{3} - R_{1}$ :

\to 100 20 - 1 30 - 2 R_{2} \leftrightarrow R_{3} 100 2 - 1 0 3 - 2 0 .

Two nonzero rows $\Rightarrow r (A) = 2$ . (The second row of the original matrix was proportional to the first, so it "vanished".)

Relation to Linear Dependence and Nonsingularity

The rank equals the maximum number of linearly independent rows (and equally of columns) of the matrix — see Linear Dependence and Independence of Rows. In particular, for a square matrix $A$ of order $n$ :

r (A) = n ⟺ det A \neq = 0 ⟺ A is nonsingular (invertible).

If instead $r (A) < n$ , the rows are linearly dependent and $det A = 0$ .