Answer to Question #232862 in Linear Algebra for PRECIOUS

Question

consider matrix A=[101 212 313 111] find the nullity and rank

Answer 1

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

$R_2=R_2-2R_1$

$$\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 3 & 1 & 3 \\ 1 & 1 & 1 \\ \end{bmatrix}$$

$R_3=R_3-3R_1$

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

$R_4=R_4-R_1$

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

$R_3=R_3-R_2$

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

$R_4=R_4-R_2$

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

The rank of a matrix is the number of nonzero rows in the reduced matrix, so the rank is $2.$

In the case of an $m\times n$ matrix, the dimension of the domain is $n,$ the number of columns in the matrix. 

By the Rank-Nullity Theorem

$${\displaystyle \operatorname {Rank} (A)+\operatorname {Nullity} (A)=n.}$$

$${Nullity} (A)=3-2=1$$

The nullity of the given matrix $A$ is $1.$

The rank of the given matrix $A$ is $2.$