To calculate:

The reduced form of the provided matrix,$\left[\begin{array}{cccc}1& 0& -3& 1\\ 0& 1& 2& 0\\ 0& 0& 3& -6\end{array}\right]$ with the use of row operations.

The reduced form of the provided matrix,

CoormaBak9
2021-02-25
Answered

To calculate:

The reduced form of the provided matrix,$\left[\begin{array}{cccc}1& 0& -3& 1\\ 0& 1& 2& 0\\ 0& 0& 3& -6\end{array}\right]$ with the use of row operations.

The reduced form of the provided matrix,

You can still ask an expert for help

Cristiano Sears

Answered 2021-02-26
Author has **96** answers

Calculation:

Consider this matrix,

$\left[\begin{array}{cccc}1& 0& -3& 1\\ 0& 1& 2& 0\\ 0& 0& 3& -6\end{array}\right]$

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.

$\left[\begin{array}{cccc}1& 0& -3& 1\\ 0& 1& 2& 0\\ 0& 0& 3& -6\end{array}\right]$

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,

$\left[\begin{array}{cccc}1& 0& -3& 1\\ 0& 1& 2& 0\\ 0& 0& 3& -6\end{array}\right]$

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,${R}_{1}\to {R}_{1}+2{R}_{3}.$

$\left[\begin{array}{cccc}1& 0& -3& 1\\ 0& 1& 2& 0\\ 0& 0& 3& -6\end{array}\right]{R}_{1}\to {R}_{1}+{R}_{3}$

$\left[\begin{array}{cccc}1& 0& 0& -5\\ 0& 1& 2& 0\\ 0& 0& 3& -6\end{array}\right]$

Similarly, use the operations${R}_{3}\to \frac{{R}_{3}}{3}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}{R}_{2}\to {R}_{2}-2\left({R}_{3}\right)$ to get the desired result as,

$\left[\begin{array}{cccc}1& 0& 0& -5\\ 0& 1& 2& 0\\ 0& 0& 3& -6\end{array}\right]{R}_{3}\to \frac{{R}_{3}}{3}$

$\left[\begin{array}{cccc}1& 0& 0& -5\\ 0& 1& 2& 0\\ 0& 0& 1& -2\end{array}\right]$

$\left[\begin{array}{cccc}1& 0& 0& -5\\ 0& 1& 2& 0\\ 0& 0& 1& -2\end{array}\right]{R}_{2}\to {R}_{2}-2{R}_{3}$

$\left[\begin{array}{cccc}1& 0& 0& -5\\ 0& 1& 0& 4\\ 0& 0& 1& -2\end{array}\right]$

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$\left[\begin{array}{cccc}1& 0& -3& 1\\ 0& 1& 2& 0\\ 0& 0& 3& -6\end{array}\right]$ is

$\left[\begin{array}{cccc}1& 0& 0& -5\\ 0& 1& 0& 4\\ 0& 0& 1& -2\end{array}\right].$

Consider this matrix,

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.

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,

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,

Similarly, use the operations

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

asked 2021-11-22

Evaluate the indefinite integral. (Use C for the constant of integration.)

$\int {e}^{\mathrm{cos}\left(9t\right)}\mathrm{sin}\left(9t\right)dt$

asked 2022-07-09

What is the polar form of $(-2,3)$ ?

asked 2022-07-08

What are the interesting parametric functions with parameters to control its nonlinearity?

$f(x)=(1-\alpha )x+\alpha g(x)$

where $0\le \alpha \le 1$

$f(x)=(1-\alpha )x+\alpha g(x)$

where $0\le \alpha \le 1$

asked 2022-05-11

Evaluate $\int -3\sqrt{-2{x}^{2}+4x+5}dx$.

asked 2020-10-28

Consider the helix represented investigation by the vector-valued function
$r(t)=\text{}\text{}2\text{}\mathrm{cos}\text{}t,\text{}2\text{}\mathrm{sin}\text{}t,\text{}t\text{}$ . Solve for t in the relationship derived in part (a), and substitute the result into the
original set of parametric equations. This yields a parametrization of the curve in terms of
the arc length parameter s.

asked 2021-08-21

Graph the sets of points whose polar coordinates satisfy the equations
and inequalities $1\le r\le 2$

asked 2021-11-21

Answer true or false to each of the following statement and explain your answer.

It is not always possible to fnd a power transformation of the response variable or the predictor variable (or both) that will straighten the scatterplot.

It is not always possible to fnd a power transformation of the response variable or the predictor variable (or both) that will straighten the scatterplot.