# 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

Reduce the following matrices to row echelon form and row reduced echelon forms:
$\left(i\right)\left[\begin{array}{ccc}1& p& -1\\ 2& 1& 7\\ -3& 3& 2\end{array}\right]\left(ii\right)\left[\begin{array}{cccc}p& -1& 7& 2\\ 2& 1& -5& 3\\ 1& 3& 2& 0\end{array}\right]$
*Find also the ra
of these matrices.
*Notice: $p=4$

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mhalmantus
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 ${R}_{2}\to {R}_{2}-2\cdot {R}_{1}$
$⇒\left[\begin{array}{ccc}1& 4& -1\\ 0& -7& 9\\ -3& 3& 2\end{array}\right]$ Apply ${R}_{3}\to {R}_{3}+3\cdot {R}_{1}$
$⇒\left[\begin{array}{ccc}1& 4& -1\\ 0& -7& 9\\ 0& 15& -1\end{array}\right]$
Step 2
Apply ${R}_{3}\to {R}_{3}+\frac{15}{7}{R}_{2}$
$⇒\left[\begin{array}{ccc}1& 4& -1\\ 0& -7& 9\\ 0& 0& \frac{128}{7}\end{array}\right]$
Apply ${R}_{2}\to -\frac{1}{7}\cdot {R}_{2}$
$⇒\left[\begin{array}{ccc}1& 4& -1\\ 0& 1& -\frac{7}{9}\\ 0& 0& \frac{128}{7}\end{array}\right]$
Step 3
Apply ${R}_{3}\to \frac{7}{128}\cdot {R}_{3}$
$⇒\left[\begin{array}{ccc}1& 4& -1\\ 0& 1& -\frac{7}{9}\\ 0& 0& 1\end{array}\right]$
This is the Row Echelon Form of the matrix.
Also, the rank of this matrix A is 3.
Apply ${R}_{1}\to {R}_{1}-4{R}_{2}$
$⇒\left[\begin{array}{ccc}1& 0& \frac{19}{9}\\ 0& 1& -\frac{7}{9}\\ 0& 0& 1\end{array}\right]$
Step 4
Apply ${R}_{1}\to {R}_{1}-\frac{19}{9}{R}_{3}$
$⇒\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& -\frac{7}{9}\\ 0& 0& 1\end{array}\right]$
Apply ${R}_{2}\to {R}_{2}+\frac{7}{9}{R}_{3}$
$⇒\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
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 ${R}_{1}↔{R}_{3}$
$⇒\left[\begin{array}{cccc}1& 3& 2& 0\\ 2& 1& -5& 3\\ 4& -1& 7& 2\end{array}\right]$
Apply ${R}_{2}\to {R}_{2}-2{R}_{1}$
$⇒\left[\begin{array}{cccc}1& 3& 2& 0\\ 0& -5& -9& 3\\ 4& -1& 7& 2\end{array}\right]$
Step 6
Apply ${R}_{3}\to {R}_{3}-4{R}_{1}$
$⇒\left[\begin{array}{cccc}1& 3& 2& 0\\ 0& -5& -9& 3\\ 0& -13& -1& 2\end{array}\right]$
Apply ${R}_{2}\to \frac{-1}{5}{R}_{2}$
Jeffrey Jordon