Ranks and Determinants

The Rank of a Matrix

The rank of an $n \times p$ matrix $A$, denoted by $\text{rank}(A)$, is the largest number of columns of $A$, which are not linearly dependent (i.e. the number of linearly independent columns).

Details

Vectors $a_1, a_2, \ldots, a_n$ are said to be linearly dependent if there exist constants $k_1, \ldots, k_n$ that are not all zero, such that

$k_1 a_1 + k_2 a_2 + \ldots + k_n a_n = 0$

Note that if such constants exist, then we can write one of the $a$ 's as a linear combination of the rest, e.g. if $k_1 \neq 0$ then

$a_1=\mathbf{c_1} = -\displaystyle\frac{k_2}{k_1} a_2 - \ldots - \displaystyle\frac{k_2}{k_1} a_n$

It can be shown that the rank of $A$, is the same as the rank of $A'$

i.e. the maximum number of linearly independent rows of $A$.

Note

Note that if $\text{rank}(A) = p$, then the columns are linearly independent.

Examples

Example

If

$$A = \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right]$$

then $\text{rank}(A)$ = 2, since

$$k_1 \left( \begin{array}{cc} 1 \\ 0 \end{array} \right) + k_2 \left( \begin{array}{cc} 0 \\ 1 \end{array} \right) = \left( \begin{array}{cc} 0 \\ 0 \end{array} \right)$$

if and only if

$$\left( \begin{array}{cc} k_1 \\ k_2 \end{array} \right) = \left( \begin{array}{cc} 0 \\ 0 \end{array} \right)$$

so the columns are linearly independent.

Example

If

$$A = \left[ \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{array} \right]$$

then $\text{rank}(A)$ = 2.

Example

If

$$A = \left[ \begin{array}{ccc} 1 & 1 & 1 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \end{array} \right]$$

then $\text{rank}(A)$ = 2. since

$$1 \left( \begin{array}{ccc} 1 \\ 0 \\ 0 \end{array} \right) + 0 \left( \begin{array}{ccc} 0 \\ 1 \\ 1 \end{array} \right) + (-1) \left( \begin{array}{ccc} 1 \\ 0 \\ 0 \end{array} \right) = 0$$

(and hence the rank cannot be more than 2) but

$$k_1 \left( \begin{array}{ccc} 1 \\ 0 \\ 0 \end{array} \right) + k_2 \left( \begin{array}{ccc} 0 \\ 1 \\ 1 \end{array} \right)$$

if and only if $k_1=k_2=0$ (and hence the rank must be at least 2).

The Determinant

Recall that for a $2 \times 2$ matrix,

$$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$

the inverse of $A$ is

$$A^{-1} = \displaystyle\frac{1}{ad-bc} \begin{bmatrix} 2 & 3 \\ 3 & 1 \end{bmatrix}$$

Details

Definition

The number $ad-bc$ is called the determinant of the $2 \times 2$ matrix $A$.

Definition

Now suppose $A$ is an $n \times n$ matrix. An elementary product from the matrix is a product of $n$ terms based on taking exactly one term from each column of row $x$. Each such term can be written in the form $a_{1j_1} \cdot a_{2j_2} \cdot a_{3j_3} \cdot \ldots \cdot a_{nj_n}$ where $j_1, \ldots, j_n$ is a permutation of the integers $1,2, \ldots, n$. Each permutation $\sigma$ of the integers $1,2,\ldots,n$ can be performed by repeatedly interchanging two numbers.

Definition

A signed elementary product is an elementary product with a positive sign if the number of interchanges in the permutation is even but negative otherwise.

The determinant of $A$, $\det(A)$ or $\vert A \vert$, is the sum of all signed elementary products.

Examples

Example

$$A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$$

then

$\vert A \vert = a_{1\underline{1}} a_{2\underline{2}} - a_{1\underline{2}}a_{2\underline{1}}$.

Example

If

$$A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}$$

Then $\vert A \vert$

= $a_{11} a_{22} a_{33}$ This is the identity permutation and has positive sign

$-a_{11} a_{23} a_{32}$ This is the permutation that only interchanges $2$ and $3$

$-a_{12} a_{21} a_{33}$ Only one interchange

$+a_{12} a_{23} a_{31}$ Two interchanges

$+a_{13} a_{21} a_{32}$ Two interchanges

$-a_{13} a_{22} a_{31}$ Three interchanges

Example

$$A = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$$

$\vert A \vert = -1$

Example

$$A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$$

$\vert A \vert = 1 \cdot 2 \cdot 3 = 6$

Example

$$A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 3 & 0 \end{bmatrix}$$

$\vert A \vert = 0$

Example

$$A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 2 \\ 0 & 3 & 0 \end{bmatrix}$$

$\vert A \vert = -6$

Example

$$A = \begin{bmatrix} 2 & 1 \\ 2 & 1 \end{bmatrix}$$

$\vert A \vert = 0$

Example

$$A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 2 \end{bmatrix}$$

$\vert A \vert = 0$

Ranks, Inverses and Determinants

The following statements are true for an $n\times n$ matrix $A$ :

• $\text{rank} (A)= n$

• $\det(A)\neq 0$

• $A$ has an inverse

Details

Suppose $A$ is an $n\times n$ matrix. Then the following are truths:

• $\text{rank} (A)= n$

• $\det(A)\neq 0$

• $A$ has an inverse