Rank of Matrix and Pivots | Linear Algebra

This lesson is a sample from The Ultimate Crash Course for PLEM Majors series.

Study linear algebra, differential equations, calculus, physics, and engineering mathematics with over 1,000 ad-free STEM lessons.

Fourthwall Store | Payhip Store

AuthorJonathanDavid.com

Rank of matrix and pivots

$\begin{bmatrix}1&1\end{bmatrix},\qquad rank(A_1)=1$

$\begin{bmatrix}1\\1\end{bmatrix},\qquad rank(A_2)=1$

$\begin{bmatrix}1&1&1\end{bmatrix},\qquad rank(A_3)=1$

$\begin{bmatrix}1\\1\\1\end{bmatrix},\qquad rank(A_4)=1$

$\begin{bmatrix}1&0\\0&1\end{bmatrix},\qquad rank(A_5)=2$

$\begin{bmatrix}1&0&0\\0&1&1\end{bmatrix},\qquad rank(A_6)=2$

$\begin{bmatrix}1&0\\0&0\\0&1\end{bmatrix},\qquad rank(A_7)=2$

$\begin{bmatrix}1&1\end{bmatrix},\qquad rank(A_8)=1$

$\begin{bmatrix}1\\0\end{bmatrix},\qquad rank(A_9)=1$

$\begin{bmatrix}1&1&1\end{bmatrix},\qquad rank(A_{10})=1$

$\begin{bmatrix}1\\0\\0\end{bmatrix},\qquad rank(A_{11})=1$

$\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix},\qquad rank(A_{12})=1$

$\begin{bmatrix}1&1&1\\1&1&-1\\1&1&1\end{bmatrix},\qquad rank(A_{13})=2$

$\begin{bmatrix}1&1&1\\0&1&1\\0&0&1\end{bmatrix},\qquad rank(A_{14})=3$

Note: max rank is the smaller dimension of $n\times m$ e.g. $3\times7$ means that 3 is the highest possible rank. It goes with the transpose as well i.e. $7\times3$ still has a highest rank of 3.

$A= \begin{bmatrix} 1&2&1&1&1&1\\ -1&-2&1&1&1&1 \end{bmatrix} \qquad R1+R2\Leftarrow R2$

$\sim \begin{bmatrix} 1&2&1&1&1&1\\ 0&0&2&2&2&2 \end{bmatrix} \Rightarrow rank(A)=2$

$Ax=b \Rightarrow \left[ \begin{array}{ccc|c} 3&2&3&1\\ 1&3&3&3\\ 3&2&1&1 \end{array} \right]$

$\sim \left[ \begin{array}{ccc|c} 1&0&0&-\frac37\\ 0&1&0&\frac87\\ 0&0&1&0 \end{array} \right], \qquad rank(A)=3$

i.e. $A=$ full rank

Continue the Linear Algebra Lesson Library

The P.L.E.M. Academy crash course system combines mathematics, physics, engineering, scientific communication, WordPress development, and STEM career preparation into one structured educational library.

Over 1,000 ad-free lessons are included through the complete crash course ecosystem.

Fourthwall Store | Payhip Store