Question Number 116884 by zakirullah last updated on 07/Oct/20
Answered by Olaf last updated on 07/Oct/20
![In linear algebra, a matrix is said to be scaled in rows if the number of zeros preceding the first non−zero value of a row increases row by row until ultimately all that remains are zero : [(⊕,∗,∗,∗,∗,∗,∗,∗,∗),(0,0,⊕,∗,∗,∗,∗,∗,∗),(0,0,0,⊕,∗,∗,∗,∗,∗),(0,0,0,0,0,0,⊕,∗,∗),(0,0,0,0,0,0,0,0,⊕),(0,0,0,0,0,0,0,0,0) ] A scaled matrix is called a reduced scaled matrix or canonical row matrix if the pivots are equal to 1 and if the other coefficients in the colums of the pivots are zero. Example : [(0,1,(13),0,(−19),0,(99)),(0,0,0,1,(12),0,(20)),(0,0,0,0,0,1,(47)),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0),(0,0,0,0,0,0,0) ]](Q116904.png)
$$\mathrm{In}\:\mathrm{linear}\:\mathrm{algebra},\:\mathrm{a}\:\mathrm{matrix}\:\mathrm{is}\:\mathrm{said}\:\mathrm{to}\:\mathrm{be} \\ $$$$\mathrm{scaled}\:\mathrm{in}\:\mathrm{rows}\:\mathrm{if}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{zeros} \\ $$$$\mathrm{preceding}\:\mathrm{the}\:\mathrm{first}\:\mathrm{non}−\mathrm{zero}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{a}\:\mathrm{row}\:\mathrm{increases}\:\mathrm{row}\:\mathrm{by}\:\mathrm{row}\:\mathrm{until} \\ $$$$\mathrm{ultimately}\:\mathrm{all}\:\mathrm{that}\:\mathrm{remains}\:\mathrm{are}\:\mathrm{zero}\:: \\ $$$$ \\ $$$$\begin{bmatrix}{\oplus}&{\ast}&{\ast}&{\ast}&{\ast}&{\ast}&{\ast}&{\ast}&{\ast}\\{\mathrm{0}}&{\mathrm{0}}&{\oplus}&{\ast}&{\ast}&{\ast}&{\ast}&{\ast}&{\ast}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\oplus}&{\ast}&{\ast}&{\ast}&{\ast}&{\ast}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\oplus}&{\ast}&{\ast}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\oplus}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}\end{bmatrix} \\ $$$$ \\ $$$$\mathrm{A}\:\mathrm{scaled}\:\mathrm{matrix}\:\mathrm{is}\:\mathrm{called}\:\mathrm{a}\:\mathrm{reduced} \\ $$$$\mathrm{scaled}\:\mathrm{matrix}\:\mathrm{or}\:\mathrm{canonical}\:\mathrm{row}\:\mathrm{matrix} \\ $$$$\mathrm{if}\:\mathrm{the}\:\mathrm{pivots}\:\mathrm{are}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{1}\:\mathrm{and}\:\mathrm{if}\:\mathrm{the} \\ $$$$\mathrm{other}\:\mathrm{coefficients}\:\mathrm{in}\:\mathrm{the}\:\mathrm{colums}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{pivots}\:\mathrm{are}\:\mathrm{zero}.\:\mathrm{Example}\:: \\ $$$$ \\ $$$$\begin{bmatrix}{\mathrm{0}}&{\mathrm{1}}&{\mathrm{13}}&{\mathrm{0}}&{−\mathrm{19}}&{\mathrm{0}}&{\mathrm{99}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{1}}&{\mathrm{12}}&{\mathrm{0}}&{\mathrm{20}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{1}}&{\mathrm{47}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}\end{bmatrix} \\ $$