Reduce the following matrices to row echelon form and row reduced echelon forms:

*Find also the ra

of these matrices.

*Notice:

Kaycee Roche
2021-02-22
Answered

Reduce the following matrices to row echelon form and row reduced echelon forms:

*Find also the ra

of these matrices.

*Notice:

You can still ask an expert for help

mhalmantus

Answered 2021-02-23
Author has **106** answers

Step 1

i) Let the matrix be,$A=\left[\begin{array}{ccc}1& 4& -1\\ 2& 1& 7\\ -3& 3& 2\end{array}\right]$

Apply${R}_{2}\to {R}_{2}-2\cdot {R}_{1}$

$\Rightarrow \left[\begin{array}{ccc}1& 4& -1\\ 0& -7& 9\\ -3& 3& 2\end{array}\right]$
Apply ${R}_{3}\to {R}_{3}+3\cdot {R}_{1}$

$\Rightarrow \left[\begin{array}{ccc}1& 4& -1\\ 0& -7& 9\\ 0& 15& -1\end{array}\right]$

Step 2

Apply${R}_{3}\to {R}_{3}+\frac{15}{7}{R}_{2}$

$\Rightarrow \left[\begin{array}{ccc}1& 4& -1\\ 0& -7& 9\\ 0& 0& \frac{128}{7}\end{array}\right]$

Apply${R}_{2}\to -\frac{1}{7}\cdot {R}_{2}$

$\Rightarrow \left[\begin{array}{ccc}1& 4& -1\\ 0& 1& -\frac{7}{9}\\ 0& 0& \frac{128}{7}\end{array}\right]$

Step 3

Apply${R}_{3}\to \frac{7}{128}\cdot {R}_{3}$

$\Rightarrow \left[\begin{array}{ccc}1& 4& -1\\ 0& 1& -\frac{7}{9}\\ 0& 0& 1\end{array}\right]$

This is the Row Echelon Form of the matrix.

Also, the rank of this matrix A is 3.

Apply${R}_{1}\to {R}_{1}-4{R}_{2}$

$\Rightarrow \left[\begin{array}{ccc}1& 0& \frac{19}{9}\\ 0& 1& -\frac{7}{9}\\ 0& 0& 1\end{array}\right]$

Step 4

Apply${R}_{1}\to {R}_{1}-\frac{19}{9}{R}_{3}$

$\Rightarrow \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& -\frac{7}{9}\\ 0& 0& 1\end{array}\right]$

Apply${R}_{2}\to {R}_{2}+\frac{7}{9}{R}_{3}$

$\Rightarrow \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

This is the Reduced Row Echelon Form of the given matrix.

Step 5

ii) Let the matrix be represented as,$B=\left[\begin{array}{cccc}4& -1& 7& 2\\ 2& 1& -5& 3\\ 1& 3& 2& 0\end{array}\right]$

Apply${R}_{1}\leftrightarrow {R}_{3}$

$\Rightarrow \left[\begin{array}{cccc}1& 3& 2& 0\\ 2& 1& -5& 3\\ 4& -1& 7& 2\end{array}\right]$

Apply${R}_{2}\to {R}_{2}-2{R}_{1}$

$\Rightarrow \left[\begin{array}{cccc}1& 3& 2& 0\\ 0& -5& -9& 3\\ 4& -1& 7& 2\end{array}\right]$

Step 6

Apply${R}_{3}\to {R}_{3}-4{R}_{1}$

$\Rightarrow \left[\begin{array}{cccc}1& 3& 2& 0\\ 0& -5& -9& 3\\ 0& -13& -1& 2\end{array}\right]$

Apply${R}_{2}\to \frac{-1}{5}{R}_{2}$

i) Let the matrix be,

Apply

Step 2

Apply

Apply

Step 3

Apply

This is the Row Echelon Form of the matrix.

Also, the rank of this matrix A is 3.

Apply

Step 4

Apply

Apply

This is the Reduced Row Echelon Form of the given matrix.

Step 5

ii) Let the matrix be represented as,

Apply

Apply

Step 6

Apply

Apply

Jeffrey Jordon

Answered 2022-01-23
Author has **2027** answers

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