# To calculate: The reduced form of the provided matrix, {left[begin{matrix}{1}&{0}&-{3}&{1}{0}&{1}&{2}&{0}{0}&{0}&{3}&-{6}end{matrix}right]} with the use of row operations.

Question
Transformation properties
To calculate:
The reduced form of the provided matrix, $${\left[\begin{matrix}{1}&{0}&-{3}&{1}\\{0}&{1}&{2}&{0}\\{0}&{0}&{3}&-{6}\end{matrix}\right]}$$ with the use of row operations.

2021-02-26
Calculation:
Consider this matrix,
$${\left[\begin{matrix}{1}&{0}&-{3}&{1}\\{0}&{1}&{2}&{0}\\{0}&{0}&{3}&-{6}\end{matrix}\right]}$$
Now, for a matrix to be in the reduced form, it must have the following properies.
1. The first non-zero element in each row must be 1, which is also known as the leading entry.
2. Each leading entry is in a column to the right of the entry in the previous row of the matrix.
3. If there are any rows with all zero elements, then they must be below the rows having a non-zero element.
Make the zeros in column 1 except the leading entry in row 1 and column 1. But here, it is observed that all the elements are already zero.
$${\left[\begin{matrix}{1}&{0}&-{3}&{1}\\{0}&{1}&{2}&{0}\\{0}&{0}&{3}&-{6}\end{matrix}\right]}$$
Similarly, make the zeros in column 2 except the leading entry in row 2 and column 2.
But here, it is observed that all the elements are already zero,
$${\left[\begin{matrix}{1}&{0}&-{3}&{1}\\{0}&{1}&{2}&{0}\\{0}&{0}&{3}&-{6}\end{matrix}\right]}$$
Then make the zeros in row 1 column 3 and row 2 column 3, except the leading entry in row 3 and column 3 with the of the operation,$${R}_{{1}}\to{R}_{{1}}+{2}{R}_{{3}}.$$
$${\left[\begin{matrix}{1}&{0}&-{3}&{1}\\{0}&{1}&{2}&{0}\\{0}&{0}&{3}&-{6}\end{matrix}\right]}{R}_{{1}}\to{R}_{{1}}+{R}_{{3}}$$
$${\left[\begin{matrix}{1}&{0}&{0}&-{5}\\{0}&{1}&{2}&{0}\\{0}&{0}&{3}&-{6}\end{matrix}\right]}$$
Similarly, use the operations $${R}_{{3}}\to\frac{{{R}_{{3}}}}{{3}}{\quad\text{and}\quad}{R}_{{2}}\to{R}_{{2}}-{2}{\left({R}_{{3}}\right)}$$ to get the desired result as,
$${\left[\begin{matrix}{1}&{0}&{0}&-{5}\\{0}&{1}&{2}&{0}\\{0}&{0}&{3}&-{6}\end{matrix}\right]}{R}_{{3}}\to\frac{{{R}_{{3}}}}{{3}}$$
$${\left[\begin{matrix}{1}&{0}&{0}&-{5}\\{0}&{1}&{2}&{0}\\{0}&{0}&{1}&-{2}\end{matrix}\right]}$$
$${\left[\begin{matrix}{1}&{0}&{0}&-{5}\\{0}&{1}&{2}&{0}\\{0}&{0}&{1}&-{2}\end{matrix}\right]}{R}_{{2}}\to{R}_{{2}}-{2}{R}_{{3}}$$
$${\left[\begin{matrix}{1}&{0}&{0}&-{5}\\{0}&{1}&{0}&{4}\\{0}&{0}&{1}&-{2}\end{matrix}\right]}$$
The matrix obtained above by performing certain row transformations satisfies all the properties of the reduced form of a matrix.
Hence, the reduced form of the matrix $${\left[\begin{matrix}{1}&{0}&-{3}&{1}\\{0}&{1}&{2}&{0}\\{0}&{0}&{3}&-{6}\end{matrix}\right]}$$ is
$${\left[\begin{matrix}{1}&{0}&{0}&-{5}\\{0}&{1}&{0}&{4}\\{0}&{0}&{1}&-{2}\end{matrix}\right]}.$$

### Relevant Questions

To calculate : The reduced form of the provided matrix, $${\left[\begin{matrix}{1}&{0}&{4}&{0}\\{0}&{1}&-{3}&-{1}\\{0}&{0}&-{2}&{2}\end{matrix}\right]}$$ with the use of row operations.
To sketch:
(i) The properties,
(ii) Linear transformation.
Let $${T}:\mathbb{R}^{2}\to\mathbb{R}^{2}$$ be the linear transformation that reflects each point through the
$$x_{1} axis.$$
Let $${A}={\left[\begin{matrix}{1}&{0}\\{0}&-{1}\end{matrix}\right]}$$
Prove that the reduced row echelon forms of the matrices
$$\displaystyle{\left({b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}\right\rbrace}{1}&{a}\mp,\ {1}&{a}\mp,\ {4}&{a}\mp,\ {8}&{a}\mp,\ {0}&{a}\mp,\ -{1}&{a}\mp,\ -{1}&\backslash{1}&{a}\mp,\ {2}&{a}\mp,\ {3}&{a}\mp,\ {9}&{a}\mp,\ {0}&{a}\mp,\ -{5}&{a}\mp,\ -{2}\backslash{0}&{a}\mp,\ -{2}&{a}\mp,\ {2}&{a}\mp,\ -{2}&{a}\mp,\ {1}&{a}\mp,\ {14}&{a}\mp,\ {3}\backslash{1}&{a}\mp,\ {4}&{a}\mp,\ {1}&{a}\mp,\ {11}&{a}\mp,\ {0}&{a}\mp,\ -{13}&{a}\mp,\ -{4}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\right)}\ {\left({b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}\right\rbrace}{0}&{a}\mp,\ -{3}&{a}\mp,\ {3}&{a}\mp,\ {1}&{a}\mp,\ {5}\backslash{0}&{a}\mp,\ {1}&{a}\mp,\ -{1}&{a}\mp,\ {0}&{a}\mp,\ {0}\backslash{0}&{a}\mp,\ {2}&{a}\mp,\ -{2}&{a}\mp,\ {0}&{a}\mp,\ -{3}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\right)}$$
are the two matrices.
The point of the reduced row echelon form is that the corresponding system of linear equations is in a particularly simple form, from which the solutions to the system $$\displaystyle{A}{X}={C}\ \in\ {\left({4}\right)}$$ can be determined immediately.
Determine if a linear transformation function is
$${T}:{M}_{{3.3}},$$
$${T}{\left({A}\right)}={\left[\begin{matrix}{0}&{0}&{1}\\{0}&{1}&{0}\\{1}&{0}&{0}\end{matrix}\right]}{A}$$
The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system of linear equations is consistent and, if so, find its general solution. Write the solution in vector form. $$\displaystyle{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&{0}&−{1}&{3}&{9}\backslash{0}&{1}&{2}&−{5}&{8}\backslash{0}&{0}&{0}&{0}&{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}$$
A small grocer finds that the monthly sales y (in $) can be approximated as a function of the amount spent advertising on the radio $$x_1$$ (in$) and the amount spent advertising in the newspaper $$x_2$$ (in $) according to $$y=ax_1+bx_2+c$$ The table gives the amounts spent in advertising and the corresponding monthly sales for 3 months. $$\begin{array}{|c|c|c|}\hline \text { Advertising, } x_{1} & \text { Advertising, } x_{2} &\text{sales, y} \\ \hline 2400 & { 800} & { 36,000} \\ \hline 2000 & { 500} & { 30,000} \\ \hline 3000 & { 1000} & { 44,000} \\ \hline\end{array}$$ a) Use the data to write a system of linear equations to solve for a, b, and c. b) Use a graphing utility to find the reduced row-echelon form of the augmented matrix. c) Write the model $$y=ax_1+bx_2+c$$ d) Predict the monthly sales if the grocer spends$250 advertising on the radio and \$500 advertising in the newspaper for a given month.
Reduce the following matrices to row echelon form and row reduced echelon forms:
$$(i)\begin{bmatrix}1 & p & -1 \\ 2 & 1 & 7 \\ -3 & 3 & 2 \end{bmatrix} (ii) \begin{bmatrix}p & -1 & 7&2 \\ 2 & 1 & -5 & 3 \\ 1 & 3 & 2 & 0 \end{bmatrix}$$

*Find also the ra
of these matrices.
*Notice: p=4
The row echelon form of a system of linear equations is given.
(a) Write the system of equations corresponding to the given matrix.
Use x, y, or x, y, z, or $$x_1,x_2,x_3, x_4$$
(b) Determine whether the system is consistent. If it is consistent, give the solution.
$$\begin{matrix}1 & 0 & 3 & 0 &1 \\ 0 & 1 & 4 & 3&2\\0&0&1&2&3\\0&0&0&0&0 \end{matrix}$$
Provide answers to all tasks using the information provided.
a) Find the parent function f.
Given Information: $$g{{\left({x}\right)}}=-{2}{\left|{x}-{1}\right|}-{4}$$
b) Find the sequence of transformation from f to g.
Given information: $$f{{\left({x}\right)}}={\left[{x}\right]}$$
c) To sketch the graph of g.
Given information: $$g{{\left({x}\right)}}=-{2}{\left|{x}-{1}\right|}-{4}$$
d) To write g in terms of f.
Given information: $$g{{\left({x}\right)}}=-{2}{\left|{x}-{1}\right|}-{4}{\quad\text{and}\quad} f{{\left({x}\right)}}={\left[{x}\right]}$$
Use x, y, or x, y, z, or $$x_1,x_2,x_3, x_4$$
$$\begin{matrix}1 & 0 & 2 & -1 \\ 0 & 1 & -4 & -2\\0&0&0&0&0 \end{matrix}$$